74:
32:
48:
17:
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401:
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The concept of strong convexity extends and parametrizes the notion of strict convexity. Intuitively, a strongly-convex function is a function that grows as fast as a quadratic function. A strongly convex function is also strictly convex, but not vice versa. If a one-dimensional function
4846:
3173:
Many properties of convex functions have the same simple formulation for functions of many variables as for functions of one variable. See below the properties for the case of many variables, as some of them are not listed for functions of one variable.
3029:. This function is not strictly convex because any two points sharing an x coordinate will have a straight line between them, while any two points NOT sharing an x coordinate will have a greater value of the function than the points between them.
406:
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1941:
1423:
723:
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407:
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Strongly convex functions are in general easier to work with than convex or strictly convex functions, since they are a smaller class. Like strictly convex functions, strongly convex functions have unique minima on compact sets.
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4654:
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9104:
9008:
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5175:{\displaystyle {\begin{aligned}f(a)+f(b)&=f\left((a+b){\frac {a}{a+b}}\right)+f\left((a+b){\frac {b}{a+b}}\right)\\&\leq {\frac {a}{a+b}}f(a+b)+{\frac {b}{a+b}}f(a+b)\\&=f(a+b).\\\end{aligned}}}
5711:
2441:
2065:
2003:
1712:
1650:
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360:. Even in infinite-dimensional spaces, under suitable additional hypotheses, convex functions continue to satisfy such properties and as a result, they are the most well-understood functionals in the
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475:
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2734:
2531:
1246:
586:
10184:
9999:
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2774:
1286:
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3451:
3378:
3027:
2467:
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4338:
4203:
540:
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9798:
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7285:
2091:
1804:
12512:
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2688:
2204:
1200:
12128:
11595:
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11993:
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9393:
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323:
270:
11914:
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4175:
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129:
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9611:
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Altenberg, L., 2012. Resolvent positive linear operators exhibit the reduction phenomenon. Proceedings of the
National Academy of Sciences, 109(10), pp.3705-3710.
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is non-negative there; this gives a practical test for convexity. Visually, a twice differentiable convex function "curves up", without any bends the other way (
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1761:
1588:
1145:
888:
12696:
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8652:
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7487:
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7212:
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6135:
6115:
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5270:
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4383:
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4313:
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4012:
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The difference of this second condition with respect to the first condition above is that this condition does not include the intersection points (for example,
1165:
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1102:
1028:
1008:
988:
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948:
928:
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433:
343:
290:
233:
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It is not necessary for a function to be differentiable in order to be strongly convex. A third definition for a strongly convex function, with parameter
10429:
13652:
5637:
11386:
is a convex function. It is strictly convex, even though the second derivative is not strictly positive at all points. It is not strongly convex.
2975:
except for the intersection points between the straight line and the curve. An example of a function which is convex but not strictly convex is
13000:
11098:
is a function that is non-negative and vanishes only at 0. This is a generalization of the concept of strongly convex function; by taking
6163:
10575:
385:
4841:{\displaystyle {\begin{aligned}f(tx_{1})&=f(tx_{1}+(1-t)\cdot 0)\\&\leq tf(x_{1})+(1-t)f(0)\\&\leq tf(x_{1}).\\\end{aligned}}}
7587:
6908:
13458:
13856:
2366:
13756:
2008:
1946:
1655:
1593:
5472:
A function that is marginally convex in each individual variable is not necessarily (jointly) convex. For example, the function
11101:
4180:
This property and the above property in terms of "...its derivative is monotonically non-decreasing..." are not equal since if
10193:
5739:
3898:
13848:
13627:
13415:
13143:
13107:
13080:
13050:
11650:, but it is not strongly convex since the second derivative can be arbitrarily close to zero. More generally, the function
1489:
1428:
789:
728:
7937:
6763:
352:
problems where they are distinguished by a number of convenient properties. For instance, a strictly convex function on an
13609:
7765:
7063:
2105:
13861:
13023:
5576:
7649:
3757:. In addition, the left derivative is left-continuous and the right-derivative is right-continuous. As a consequence,
2316:
13584:
13333:
13297:
13272:
13177:
7492:
2269:
11281:
is a convex function. It is also strongly convex (and hence strictly convex too), with strong convexity constant 2.
6640:
376:
is always bounded above by the expected value of the convex function of the random variable. This result, known as
348:
Convex functions play an important role in many areas of mathematics. They are especially important in the study of
9109:
6053:
3891:
A differentiable function of one variable is convex on an interval if and only if its graph lies above all of its
13881:
13798:
12931:
12896:
11702:
3141:. If the term "convex" is used without an "up" or "down" keyword, then it refers strictly to a cup shaped graph
12836:
12218:
8690:
14100:
13379:
9685:
8373:
is twice continuously differentiable and the domain is the real line, then we can characterize it as follows:
6542:
12399:
8180:
13637:
13619:
4851:
2533:
is also undefined so a convex extended real-valued function is typically only allowed to take exactly one of
10147:
Assuming still that the function is twice continuously differentiable, one can show that the lower bound of
5455:
3881:
3754:
3597:
13901:
12174:
11329:
10545:
by the assumption about the eigenvalues, and hence we recover the second strong convexity equation above.
5185:
4451:
14105:
13632:
13614:
8111:
6862:
6709:
this property extends to infinite sums, integrals and expected values as well (provided that they exist).
6594:
3885:
446:
12552:
11751:
9827:
7874:
6717:
6234:
13866:
11600:
10104:
then this means the
Hessian is positive semidefinite (or if the domain is the real line, it means that
8565:
5990:
2472:
12701:
12004:
6312:
5943:
5295:
2693:
2507:
1205:
545:
13906:
13896:
12916:
10150:
9965:
9883:
8746:
5896:
2739:
2100:
1251:
12773:
4620:
4074:
4025:). If its second derivative is positive at all points then the function is strictly convex, but the
3550:
3407:
3334:
2978:
2446:
389:
14095:
13886:
13871:
13713:
12926:
12668:
12632:
11653:
11234:
9873:
5861:
3657:(or vice versa). This characterization of convexity is quite useful to prove the following results.
513:
381:
13264:
12996:
12517:
12450:
12361:
11474:
10107:
10041:
9773:
8399:
7534:
7416:
7237:
5852:
A twice differentiable function of several variables is convex on a convex set if and only if its
2074:
1766:
13956:
13933:
13827:
13751:
12891:
12485:
12329:
12043:
11822:
10682:
8483:
3873:
3778:
2661:
2186:
1936:{\displaystyle f\left(tx_{1}+(1-t)x_{2}\right)\leq tf\left(x_{1}\right)+(1-t)f\left(x_{2}\right)}
1418:{\displaystyle f\left(tx_{1}+(1-t)x_{2}\right)\leq tf\left(x_{1}\right)+(1-t)f\left(x_{2}\right)}
1173:
718:{\displaystyle f\left(tx_{1}+(1-t)x_{2}\right)\leq tf\left(x_{1}\right)+(1-t)f\left(x_{2}\right)}
159:
13130:
12091:
11558:
11287:
11192:
9145:
8315:
8145:
6377:
4388:
4032:
2906:{\displaystyle f\left(tx_{1}+(1-t)x_{2}\right)<tf\left(x_{1}\right)+(1-t)f\left(x_{2}\right)}
14074:
14035:
13951:
13876:
13803:
13788:
13741:
13554:
13509:
13407:
13350:
12906:
10144:), which implies the function is convex, and perhaps strictly convex, but not strongly convex.
6256:
6094:
5632:
5475:
5459:
3750:
3162:
2589:
377:
361:
102:
21:
14020:
14012:
14008:
14004:
14000:
13996:
13325:
13114:
13057:
11972:
11396:
9807:
8257:
7310:
4584:
2559:
2536:
1147:
So, this condition requires that the straight line between any pair of points on the curve of
13808:
13167:
12936:
12675:
12133:
11942:
11526:
10746:
10004:
9561:
8657:
5571:
5526:
2611:
1806:
In fact, the intersection points do not need to be considered in a condition of convex using
1567:(the straight line is represented by the right hand side of this condition) and the curve of
180:
101:
lies above or on the graph between the two points. Equivalently, a function is convex if its
13399:
13317:
13256:
12274:
11854:
10894:
10722:
9616:
9357:
8850:
8793:
7698:
7008:
5447:{\displaystyle f\!\left({\frac {x_{1}+x_{2}}{2}}\right)\leq {\frac {f(x_{1})+f(x_{2})}{2}}.}
4017:
A twice differentiable function of one variable is convex on an interval if and only if its
3603:
298:
245:
13813:
13679:
13644:
13594:
12941:
11892:
11553:
11158:
11081:
10828:
9613:
Despite this, functions exist that are strictly convex but are not strongly convex for any
7733:
7346:
7002:
6263:
6018:
4153:
3633:
3503:
3476:
3144:
1060:
1033:
293:
171:
138:
114:
86:
52:
13502:
12587:
12300:
11445:
10419:{\displaystyle f(y)=f(x)+\nabla f(x)^{T}(y-x)+{\frac {1}{2}}(y-x)^{T}\nabla ^{2}f(z)(y-x)}
9590:
6830:
4124:
8:
13746:
13736:
13731:
13579:. River Edge, NJ: World Scientific Publishing Co., Inc. pp. xx+367.
13486:
13257:
12881:
12832:
11439:
10868:
10187:
10081:
9331:
8876:
8289:
6490:
5826:
4318:
4183:
3704:
1717:
357:
98:
36:
12968:
11919:
9942:
9308:
8611:
8439:
7831:
6424:
6289:
6140:
5870:
4273:
4228:
3710:
3069:
1743:
1570:
1127:
870:
13975:
13693:
12886:
12847:
12828:
12681:
12085:
11998:
11731:
11708:
10848:
10662:
10642:
10551:
9922:
9753:
9733:
9665:
9645:
9169:
8830:
8637:
8545:
8523:
8463:
8379:
8356:
8091:
7917:
7854:
7472:
7396:
7376:
7290:
7217:
7197:
7177:
7149:
7129:
7043:
6516:
6470:
6447:
6269:
6120:
6100:
6033:
5857:
5719:
5275:
5255:
4564:
4536:
4516:
4423:
4368:
4343:
4298:
4253:
4208:
3997:
3977:
3848:
3828:
3808:
3788:
3760:
3732:
3686:
3663:
3530:
3456:
3401:
3383:
3186:
3092:
3035:
2958:
2938:
2918:
2635:
2249:
2229:
2209:
2166:
1550:
1150:
1107:
1087:
1013:
993:
973:
953:
933:
913:
893:
850:
483:
418:
365:
328:
275:
240:
218:
10749:, which states that a continuous function on a compact set has a maximum and minimum.
8078:{\displaystyle \left\{(x,t):{\tfrac {x}{t}}\in \operatorname {Dom} (f),t>0\right\}}
2071:
The second statement characterizing convex functions that are valued in the real line
13793:
13580:
13475:
13411:
13400:
13375:
13329:
13318:
13293:
13268:
13231:
13173:
13139:
13103:
13076:
13046:
13019:
12920:
12911:
9549:{\displaystyle f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)-{\frac {1}{2}}mt(1-t)\|x-y\|_{2}^{2}}
9173:
6856:
is the collection of points where the expression is finite. Important special cases:
4018:
167:
13946:
13891:
13782:
13777:
13542:
13530:
13440:
12871:
12861:
11966:
4026:
4022:
3130:
3055:
132:
13489:, and Lewis, Adrian. (2000). Convex Analysis and Nonlinear Optimization. Springer.
5454:
This condition is only slightly weaker than convexity. For example, a real-valued
13966:
13937:
13911:
13832:
13726:
13675:
13590:
13263:. Contributors: Angelia Nedic and Asuman E. Ozdaglar. Athena Scientific. p.
13097:
13070:
13040:
12866:
12843:
12767:
11175:
to be an increasing function, but this condition is not required by all authors.
4441:
373:
175:
9662:
is twice continuously differentiable, then it is strongly convex with parameter
9296:{\displaystyle f(y)\geq f(x)+\nabla f(x)^{T}(y-x)+{\frac {m}{2}}\|y-x\|_{2}^{2}}
14053:
13961:
13822:
13721:
12901:
11390:
10809:
9801:
6252:
6084:
5853:
4445:
369:
163:
9558:
Notice that this definition approaches the definition for strict convexity as
6255:. Indeed, convex functions are exactly those that satisfies the hypothesis of
5462:. In particular, a continuous function that is midpoint convex will be convex.
14089:
13837:
13444:
13201:"If f is strictly convex in a convex set, show it has no more than 1 minimum"
12946:
10798:
9139:
6080:
4581:
is convex, by using one of the convex function definitions above and letting
3884:
on that interval. If a function is differentiable and convex then it is also
3681:
2935:
is a function that the straight line between any pair of points on the curve
1547:) between the straight line passing through a pair of points on the curve of
9099:{\displaystyle \langle \nabla f(x)-\nabla f(y),x-y\rangle \geq m\|x-y\|^{2}}
14058:
12876:
5891:
5517:
440:
349:
94:
13200:
9172:. Some authors, such as refer to functions satisfying this inequality as
47:
14048:
14043:
13927:
13459:
Convexity of the dominant eigenvalue of an essentially nonnegative matrix
5520:, and thus marginally convex, in each variable, but not (jointly) convex.
3204:
236:
174:. Well-known examples of convex functions of a single variable include a
82:
11725:
is a convex function. The term "superconvex" is sometimes used instead.
73:
13971:
13941:
13703:
13508:
13431:
Kingman, J. F. C. (1961). "A Convexity
Property of Positive Matrices".
13042:
Methods of
Mathematics Applied to Calculus, Probability, and Statistics
9877:
9003:{\displaystyle (\nabla f(x)-\nabla f(y))^{T}(x-y)\geq m\|x-y\|_{2}^{2}}
3877:
3324:{\displaystyle R(x_{1},x_{2})={\frac {f(x_{2})-f(x_{1})}{x_{2}-x_{1}}}}
3165:
refers to an inequality involving a convex or convex-(down), function.
436:
108:
40:
13461:. Proceedings of the American Mathematical Society, 81(4), pp.657-658.
6412:) that is convex in one variable must be convex in the other variable.
6224:{\displaystyle \operatorname {E} (f(X))\geq f(\operatorname {E} (X)),}
12623:
11071:{\displaystyle f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)-t(1-t)\phi (\|x-y\|)}
10778:
6015:
are convex sets. A function that satisfies this property is called a
3876:
function of one variable is convex on an interval if and only if its
3782:
35:
A function (in black) is convex if and only if the region above its
28:
Real function with secant line between points above the graph itself
13980:
7639:{\displaystyle A\in \mathbf {R} ^{m\times n},b\in \mathbf {R} ^{m}}
6993:{\displaystyle g(x)=\max \left\{f_{1}(x),\ldots ,f_{n}(x)\right\}.}
3380:
is the slope of the purple line in the first drawing; the function
2582:
The second statement can also be modified to get the definition of
1590:
the first condition includes the intersection points as it becomes
353:
31:
13761:
13102:(illustrated ed.). Cambridge University Press. p. 160.
10745:
is strongly convex. The proof of this statement follows from the
10538:{\displaystyle (y-x)^{T}\nabla ^{2}f(z)(y-x)\geq m(y-x)^{T}(y-x)}
6027:
3892:
111:. In simple terms, a convex function graph is shaped like a cup
16:
12763:. This statement also holds if we replace "convex" by "concave".
506:
if and only if any of the following equivalent conditions hold:
13397:
2206:
as a value. The first statement is not used because it permits
107:(the set of points on or above the graph of the function) is a
13402:
Convex
Analysis and Monotone Operator Theory in Hilbert Spaces
13099:
The
Probability Companion for Engineering and Computer Science
8654:
be strictly convex, and suppose there is a sequence of points
5706:{\displaystyle \{(x,r)\in X\times \mathbb {R} ~:~r\geq f(x)\}}
2067:
are always true (so not useful to be a part of a condition).
12626:
function, also called softmax function, is a convex function.
9587:
and is identical to the definition of a convex function when
2436:{\displaystyle tf\left(x_{1}\right)+(1-t)f\left(x_{2}\right)}
13674:
13505:. (2004). Fundamentals of Convex analysis. Berlin: Springer.
13320:
Introductory
Lectures on Convex Optimization: A Basic Course
7413:
is convex and non-increasing over a univariate domain, then
2060:{\displaystyle f\left(x_{2}\right)\leq f\left(x_{2}\right)}
1998:{\displaystyle f\left(x_{1}\right)\leq f\left(x_{1}\right)}
1707:{\displaystyle f\left(x_{2}\right)\leq f\left(x_{2}\right)}
1645:{\displaystyle f\left(x_{1}\right)\leq f\left(x_{1}\right)}
1030:
in the left hand side represents the straight line between
13045:(illustrated ed.). Courier Corporation. p. 227.
11442:), even though it does not have a derivative at the point
10756:
1010:
sweeps this line. Similarly, the argument of the function
13290:
Introduction to numerical linear algebra and optimisation
11155:
It is worth noting that some authors require the modulus
11145:{\displaystyle \phi (\alpha )={\tfrac {m}{2}}\alpha ^{2}}
6706:
In particular, the sum of two convex functions is convex.
725:
The right hand side represents the straight line between
5816:{\displaystyle f(x)\geq f(y)+\nabla f(y)^{T}\cdot (x-y)}
5458:
that is midpoint-convex is convex: this is a theorem of
13348:
12419:
12243:
12194:
11121:
8722:
8029:
7973:
7469:
Convexity is invariant under affine maps: that is, if
1540:{\displaystyle \left(x_{2},f\left(x_{2}\right)\right)}
1479:{\displaystyle \left(x_{1},f\left(x_{1}\right)\right)}
840:{\displaystyle \left(x_{2},f\left(x_{2}\right)\right)}
779:{\displaystyle \left(x_{1},f\left(x_{1}\right)\right)}
12776:
12704:
12684:
12635:
12590:
12555:
12520:
12488:
12453:
12402:
12364:
12332:
12303:
12277:
12221:
12177:
12136:
12094:
12046:
12007:
11975:
11945:
11922:
11895:
11857:
11825:
11754:
11734:
11711:
11656:
11603:
11561:
11529:
11477:
11448:
11399:
11332:
11290:
11237:
11195:
11161:
11104:
11084:
10938:
10897:
10871:
10851:
10831:
10725:
10685:
10665:
10645:
10629:{\displaystyle x\mapsto f(x)-{\frac {m}{2}}\|x\|^{2}}
10578:
10554:
10432:
10276:
10196:
10153:
10110:
10084:
10044:
10007:
9968:
9945:
9925:
9886:
9830:
9810:
9776:
9756:
9736:
9688:
9668:
9648:
9619:
9593:
9564:
9400:
9360:
9334:
9311:
9185:
9148:
9112:
9016:
8905:
8879:
8853:
8833:
8796:
8749:
8693:
8660:
8640:
8614:
8568:
8548:
8526:
8486:
8466:
8442:
8402:
8382:
8359:
8318:
8292:
8260:
8183:
8148:
8114:
8094:
8004:
7991:{\displaystyle g(x,t)=tf\left({\tfrac {x}{t}}\right)}
7940:
7920:
7877:
7857:
7834:
7768:
7736:
7701:
7652:
7590:
7537:
7495:
7475:
7419:
7399:
7379:
7349:
7313:
7293:
7240:
7220:
7200:
7180:
7152:
7132:
7066:
7046:
7011:
6911:
6865:
6833:
6820:{\displaystyle g(x)=\sup \nolimits _{i\in I}f_{i}(x)}
6766:
6720:
6643:
6597:
6545:
6519:
6493:
6473:
6450:
6427:
6416:
6380:
6315:
6292:
6272:
6237:
6166:
6143:
6123:
6103:
6091:
convex function will have at most one global minimum.
6056:
6036:
5993:
5946:
5899:
5873:
5829:
5742:
5722:
5640:
5579:
5529:
5478:
5343:
5298:
5278:
5258:
5188:
4918:
4854:
4657:
4623:
4587:
4567:
4539:
4519:
4454:
4426:
4391:
4371:
4346:
4321:
4301:
4276:
4256:
4231:
4211:
4186:
4156:
4127:
4077:
4035:
4029:
does not hold. For example, the second derivative of
4000:
3980:
3901:
3851:
3831:
3811:
3805:
is not differentiable can however still be dense. If
3791:
3763:
3735:
3713:
3689:
3666:
3636:
3606:
3553:
3533:
3506:
3479:
3459:
3410:
3386:
3337:
3213:
3189:
3147:
3109:
multiplied by −1) is convex (resp. strictly convex).
3095:
3072:
3038:
2981:
2961:
2941:
2921:
2782:
2742:
2696:
2664:
2638:
2614:
2592:
2562:
2539:
2510:
2475:
2449:
2369:
2319:
2272:
2252:
2232:
2212:
2189:
2169:
2108:
2077:
2011:
1949:
1812:
1769:
1746:
1720:
1658:
1596:
1573:
1553:
1492:
1431:
1294:
1254:
1208:
1176:
1153:
1130:
1110:
1090:
1063:
1036:
1016:
996:
976:
956:
936:
916:
896:
873:
853:
792:
731:
594:
548:
516:
486:
449:
421:
411:
Visualizing a convex function and Jensen's
Inequality
331:
301:
278:
248:
221:
183:
141:
131:(or a straight line like a linear function), while a
117:
13018:(8th ed.). Cengage Learning. pp. 223–224.
6137:
is a random variable taking values in the domain of
5736:
defined on a convex domain is convex if and only if
13287:
9876:. This is equivalent to requiring that the minimum
7821:{\displaystyle g(x)=\inf \nolimits _{y\in C}f(x,y)}
7119:{\displaystyle g(x)=\sup \nolimits _{y\in C}f(x,y)}
12816:
12755:
12690:
12659:
12602:
12576:
12541:
12506:
12474:
12439:
12385:
12350:
12318:
12289:
12263:
12207:
12160:
12122:
12073:
12032:
11987:
11957:
11931:
11908:
11878:
11843:
11811:
11740:
11717:
11693:
11642:
11589:
11541:
11515:
11460:
11430:
11374:
11318:
11269:
11223:
11167:
11144:
11090:
11070:
10924:
10883:
10857:
10837:
10737:
10711:
10671:
10651:
10628:
10560:
10537:
10418:
10262:
10178:
10136:
10096:
10070:
10030:
9993:
9954:
9931:
9911:
9864:
9816:
9792:
9762:
9742:
9722:
9674:
9654:
9631:
9605:
9579:
9548:
9387:
9346:
9320:
9295:
9160:
9130:
9098:
9002:
8891:
8865:
8839:
8816:
8782:
8735:
8679:
8646:
8623:
8600:
8554:
8532:
8512:
8472:
8451:
8428:
8388:
8365:
8336:
8304:
8278:
8246:
8169:
8134:
8100:
8077:
7990:
7926:
7904:
7863:
7843:
7820:
7754:
7722:
7683:
7638:
7576:
7523:
7481:
7458:
7405:
7385:
7362:
7335:
7299:
7279:
7226:
7206:
7186:
7158:
7138:
7118:
7052:
7032:
6992:
6897:
6848:
6819:
6752:
6698:
6629:
6583:
6525:
6505:
6479:
6456:
6436:
6404:
6366:
6301:
6278:
6243:
6223:
6152:
6129:
6109:
6068:
6042:
6007:
5979:
5932:
5882:
5841:
5815:
5728:
5705:
5623:
5562:
5508:
5446:
5330:
5284:
5264:
5239:
5174:
4904:
4840:
4644:
4609:
4573:
4545:
4525:
4505:
4432:
4412:
4377:
4352:
4332:
4307:
4287:
4262:
4242:
4217:
4197:
4169:
4142:
4113:
4063:
4006:
3986:
3966:
3857:
3837:
3817:
3797:
3769:
3741:
3722:
3695:
3672:
3649:
3622:
3588:
3539:
3519:
3492:
3465:
3445:
3392:
3372:
3323:
3195:
3153:
3101:
3081:
3044:
3021:
2967:
2947:
2927:
2905:
2768:
2728:
2682:
2644:
2624:
2600:
2571:
2548:
2525:
2496:
2461:
2435:
2355:
2305:
2258:
2238:
2218:
2198:
2175:
2156:{\displaystyle =\mathbb {R} \cup \{\pm \infty \},}
2155:
2085:
2059:
1997:
1935:
1798:
1755:
1732:
1706:
1644:
1582:
1559:
1539:
1478:
1417:
1280:
1240:
1194:
1159:
1139:
1116:
1096:
1076:
1049:
1022:
1002:
982:
962:
942:
922:
902:
882:
859:
839:
778:
717:
580:
534:
492:
469:
427:
337:
317:
284:
264:
227:
207:
147:
123:
13254:
11851:is convex; it is continuous on the open interval
8873:if the following inequality holds for all points
7234:is non-decreasing over a univariate domain, then
5624:{\displaystyle =\mathbb {R} \cup \{\pm \infty \}}
5466:
5347:
14087:
13128:
12645:
7684:{\displaystyle D_{g}\subseteq \mathbf {R} ^{n}.}
6927:
2443:would be undefined (because the multiplications
2356:{\displaystyle f\left(x_{2}\right)=\pm \infty ,}
13129:Boyd, Stephen P.; Vandenberghe, Lieven (2004).
11152:we recover the definition of strong convexity.
7524:{\displaystyle D_{f}\subseteq \mathbf {R} ^{m}}
4385:is a convex function of one real variable, and
2306:{\displaystyle f\left(x_{1}\right)=\pm \infty }
13315:
13095:
12850:is a convex function of its diagonal elements.
12084:Examples of functions that are convex but not
11597:is convex. It is also strictly convex, since
11183:
8790:, the function is not strongly convex because
8540:(note: this is sufficient, but not necessary).
6699:{\displaystyle w_{1}f_{1}+\cdots +w_{n}f_{n}.}
3845:may fail to be continuous at the endpoints of
13660:
10820:
10765:is a strongly-convex function with parameter
10639:A twice continuously differentiable function
9131:{\displaystyle \langle \cdot ,\cdot \rangle }
4270:while its converse is not true, for example,
13369:
13365:
13363:
12678:is convex but not strictly convex, since if
11062:
11050:
10617:
10610:
10257:
10203:
9532:
9519:
9279:
9266:
9155:
9149:
9125:
9113:
9087:
9074:
9065:
9017:
8986:
8973:
8347:
6735:
6721:
6069:{\displaystyle {\operatorname {argmin} }\,f}
5974:
5947:
5927:
5900:
5700:
5641:
5618:
5609:
3177:
2586:, where the latter is obtained by replacing
2147:
2138:
13553:
13393:
13391:
13311:
13309:
12613:
13667:
13653:
13541:
13529:
13038:
12264:{\displaystyle f''(x)={\tfrac {2}{x^{3}}}}
10825:A uniformly convex function, with modulus
10186:implies that it is strongly convex. Using
9962:If the domain is just the real line, then
9179:An equivalent condition is the following:
8827:More generally, a differentiable function
8736:{\displaystyle f''(x_{n})={\tfrac {1}{n}}}
6760:be a collection of convex functions. Then
386:arithmetic–geometric mean inequality
13628:"Convex function (of a complex variable)"
13574:
13520:
13474:
13360:
13248:
9723:{\displaystyle \nabla ^{2}f(x)\succeq mI}
8847:is called strongly convex with parameter
6584:{\displaystyle w_{1},\ldots ,w_{n}\geq 0}
6062:
6001:
5669:
5602:
3207:variable defined on an interval, and let
2615:
2597:
2593:
2131:
2079:
463:
13577:Convex analysis in general vector spaces
13561:. Princeton: Princeton University Press.
13492:
13398:H. Bauschke and P. L. Combettes (2011).
13388:
13372:Convex Analysis in General Vector Spaces
13306:
13165:
12440:{\displaystyle f(x)={\tfrac {1}{x^{2}}}}
10263:{\displaystyle z\in \{tx+(1-t)y:t\in \}}
8247:{\displaystyle f(ax+by)\leq af(x)+bf(y)}
3967:{\displaystyle f(x)\geq f(y)+f'(y)(x-y)}
399:
162:function of a single variable is convex
72:
46:
30:
15:
13757:Locally convex topological vector space
13565:
13430:
13351:"Optimization III: Convex Optimization"
13324:. Kluwer Academic Publishers. pp.
13281:
13232:"Strong convexity · Xingyu Zhou's blog"
13013:
10757:Properties of strongly-convex functions
7370:is convex and monotonically increasing.
4905:{\displaystyle f(tx_{1})\leq tf(x_{1})}
97:between any two distinct points on the
14088:
13610:"Convex function (of a real variable)"
13568:Symmetric Properties of Real Functions
13124:
13122:
13068:
13648:
12824:is simultaneously convex and concave.
12770:, that is, each function of the form
11939:; thus it is convex on the set where
6023:and may fail to be a convex function.
3680:of one real variable defined on some
3112:
2093:is also the statement used to define
1167:to be above or just meets the graph.
292:as a nonnegative real number) and an
13547:Optimization by Vector Space Methods
13495:Distributions and Fourier Transforms
13433:The Quarterly Journal of Mathematics
13169:Distributions and Fourier Transforms
12208:{\displaystyle f(x)={\tfrac {1}{x}}}
11375:{\displaystyle f''(x)=12x^{2}\geq 0}
5240:{\displaystyle f(a)+f(b)\leq f(a+b)}
4506:{\displaystyle f(a+b)\geq f(a)+f(b)}
13501:Hiriart-Urruty, Jean-Baptiste, and
13119:
13075:. Mir Publishers. p. 126-127.
13069:Uvarov, Vasiliĭ Borisovich (1988).
11886:but not continuous at 0 and 1.
8135:{\displaystyle f:X\to \mathbf {R} }
6898:{\displaystyle f_{1},\ldots ,f_{n}}
6630:{\displaystyle f_{1},\ldots ,f_{n}}
4295:is monotonically non-decreasing on
4250:is monotonically non-decreasing on
470:{\displaystyle f:X\to \mathbb {R} }
368:, a convex function applied to the
13:
13514:Convex Functions and Orlicz Spaces
13342:
12577:{\displaystyle (-\infty ,\infty )}
12568:
12562:
12527:
12498:
12469:
12371:
12342:
11812:{\displaystyle f(0)=f(1)=1,f(x)=0}
10568:is strongly convex with parameter
10456:
10380:
10307:
10155:
9970:
9888:
9865:{\displaystyle \nabla ^{2}f(x)-mI}
9832:
9778:
9690:
9216:
9038:
9020:
8927:
8909:
8286:and any non-negative real numbers
7905:{\displaystyle g(x)\neq -\infty .}
7896:
6753:{\displaystyle \{f_{i}\}_{i\in I}}
6417:Operations that preserve convexity
6244:{\displaystyle \operatorname {E} }
6238:
6200:
6167:
5864:on the interior of the convex set.
5773:
5615:
5592:
5586:
5554:
5548:
5272:is midpoint convex on an interval
2566:
2543:
2520:
2514:
2488:
2456:
2347:
2300:
2193:
2144:
2121:
2115:
14:
14117:
13602:
13203:. Math StackExchange. 21 Mar 2013
12549:, but not convex on the interval
11643:{\displaystyle f''(x)=e^{x}>0}
8601:{\displaystyle f''(x)\geq m>0}
6097:applies to every convex function
6008:{\displaystyle a\in \mathbb {R} }
4340:is not defined at some points on
2497:{\displaystyle 0\cdot (-\infty )}
13535:Linear and Nonlinear Programming
13480:Convex Analysis and Optimization
13259:Convex Analysis and Optimization
13003:from the original on 2013-12-18.
12756:{\displaystyle f(a+b)=f(a)+f(b)}
12584:, because of the singularity at
12358:. It is concave on the interval
12033:{\displaystyle f(x)={\sqrt {x}}}
8128:
7934:is convex, then its perspective
7668:
7626:
7599:
7511:
6905:are convex functions then so is
6367:{\displaystyle f(ax,ay)=af(x,y)}
6309:(that is, a function satisfying
5980:{\displaystyle \{x:f(x)\leq a\}}
5331:{\displaystyle x_{1},x_{2}\in C}
2729:{\displaystyle x_{1},x_{2}\in X}
2526:{\displaystyle -\infty +\infty }
1241:{\displaystyle x_{1},x_{2}\in X}
581:{\displaystyle x_{1},x_{2}\in X}
13862:Ekeland's variational principle
13451:
13424:
13349:Nemirovsky and Ben-Tal (2023).
13224:
13215:
12932:Logarithmically convex function
11997:Examples of functions that are
11438:is convex (as reflected in the
10179:{\displaystyle \nabla ^{2}f(x)}
9994:{\displaystyle \nabla ^{2}f(x)}
9912:{\displaystyle \nabla ^{2}f(x)}
8824:will become arbitrarily small.
8783:{\displaystyle f''(x_{n})>0}
8562:strongly convex if and only if
6083:of a convex function is also a
5933:{\displaystyle \{x:f(x)<a\}}
4205:is non-negative on an interval
2769:{\displaystyle x_{1}\neq x_{2}}
1281:{\displaystyle x_{1}\neq x_{2}}
345:as a nonnegative real number).
13525:. World Scientific Publishing.
13292:. Cambridge University Press.
13193:
13172:. Academic Press. p. 12.
13159:
13138:. Cambridge University Press.
13089:
13062:
13032:
13007:
12989:
12961:
12817:{\displaystyle f(x)=a^{T}x+b,}
12786:
12780:
12750:
12744:
12735:
12729:
12720:
12708:
12654:
12648:
12571:
12556:
12536:
12521:
12501:
12489:
12463:
12457:
12412:
12406:
12380:
12365:
12345:
12333:
12313:
12307:
12236:
12230:
12187:
12181:
12146:
12140:
12104:
12098:
12056:
12050:
12017:
12011:
11870:
11858:
11800:
11794:
11779:
11773:
11764:
11758:
11686:
11680:
11666:
11660:
11618:
11612:
11571:
11565:
11503:
11494:
11487:
11481:
11424:
11416:
11409:
11403:
11347:
11341:
11300:
11294:
11252:
11246:
11205:
11199:
11114:
11108:
11065:
11047:
11041:
11029:
11020:
11014:
11008:
10996:
10990:
10984:
10972:
10966:
10954:
10942:
10916:
10904:
10700:
10694:
10594:
10588:
10582:
10532:
10520:
10511:
10498:
10489:
10477:
10474:
10468:
10446:
10433:
10413:
10401:
10398:
10392:
10370:
10357:
10341:
10329:
10320:
10313:
10301:
10295:
10286:
10280:
10254:
10242:
10227:
10215:
10173:
10167:
10125:
10119:
10059:
10053:
10022:
10016:
10001:is just the second derivative
9988:
9982:
9906:
9900:
9850:
9844:
9708:
9702:
9568:
9516:
9504:
9482:
9476:
9470:
9458:
9452:
9446:
9434:
9428:
9416:
9404:
9379:
9367:
9250:
9238:
9229:
9222:
9210:
9204:
9195:
9189:
9050:
9044:
9032:
9026:
8964:
8952:
8943:
8939:
8933:
8921:
8915:
8906:
8811:
8805:
8771:
8758:
8715:
8702:
8674:
8661:
8583:
8577:
8501:
8495:
8417:
8411:
8241:
8235:
8223:
8217:
8205:
8187:
8158:
8152:
8124:
8055:
8049:
8022:
8010:
7956:
7944:
7887:
7881:
7815:
7803:
7778:
7772:
7749:
7737:
7717:
7705:
7571:
7556:
7547:
7541:
7453:
7450:
7444:
7438:
7429:
7423:
7328:
7322:
7274:
7271:
7265:
7259:
7250:
7244:
7113:
7101:
7076:
7070:
7027:
7015:
6979:
6973:
6951:
6945:
6921:
6915:
6843:
6837:
6814:
6808:
6776:
6770:
6361:
6349:
6337:
6319:
6215:
6212:
6206:
6197:
6188:
6185:
6179:
6173:
5965:
5959:
5918:
5912:
5810:
5798:
5786:
5779:
5767:
5761:
5752:
5746:
5697:
5691:
5656:
5644:
5595:
5580:
5557:
5542:
5539:
5494:
5482:
5467:Functions of several variables
5432:
5419:
5410:
5397:
5234:
5222:
5213:
5207:
5198:
5192:
5162:
5150:
5134:
5122:
5095:
5083:
5026:
5014:
4977:
4965:
4947:
4941:
4932:
4926:
4899:
4886:
4874:
4858:
4828:
4815:
4796:
4790:
4784:
4772:
4766:
4753:
4734:
4725:
4713:
4694:
4681:
4665:
4645:{\displaystyle 0\leq t\leq 1,}
4500:
4494:
4485:
4479:
4470:
4458:
4401:
4395:
4114:{\displaystyle f''(x)=12x^{2}}
4092:
4086:
4045:
4039:
3961:
3949:
3946:
3940:
3926:
3920:
3911:
3905:
3589:{\displaystyle R(x_{1},x_{2})}
3583:
3557:
3473:does not change by exchanging
3446:{\displaystyle (x_{1},x_{2}),}
3437:
3411:
3373:{\displaystyle R(x_{1},x_{2})}
3367:
3341:
3290:
3277:
3268:
3255:
3243:
3217:
3022:{\displaystyle f(x,y)=x^{2}+y}
2997:
2985:
2879:
2867:
2819:
2807:
2491:
2482:
2462:{\displaystyle 0\cdot \infty }
2409:
2397:
2266:as a value, in which case, if
2124:
2109:
1909:
1897:
1849:
1837:
1391:
1379:
1331:
1319:
691:
679:
631:
619:
459:
193:
187:
135:'s graph is shaped like a cap
1:
13516:. Groningen: P.Noordhoff Ltd.
13493:Donoghue, William F. (1969).
13468:
13166:Donoghue, William F. (1969).
13096:Prügel-Bennett, Adam (2020).
12997:"Concave Upward and Downward"
12919:, which relates convexity to
12831:is a convex function, by the
12660:{\displaystyle -\log \det(X)}
11694:{\displaystyle g(x)=e^{f(x)}}
11270:{\displaystyle f''(x)=2>0}
5631:is convex if and only if its
4617:it follows that for all real
3866:
3168:
535:{\displaystyle 0\leq t\leq 1}
395:
170:is nonnegative on its entire
13288:Philippe G. Ciarlet (1989).
13039:W. Hamming, Richard (2012).
12542:{\displaystyle (-\infty ,0)}
12482:, is convex on the interval
12475:{\displaystyle f(0)=\infty }
12386:{\displaystyle (-\infty ,0)}
11516:{\displaystyle f(x)=|x|^{p}}
10572:if and only if the function
10137:{\displaystyle f''(x)\geq 0}
10071:{\displaystyle f''(x)\geq m}
9793:{\displaystyle \nabla ^{2}f}
8429:{\displaystyle f''(x)\geq 0}
7577:{\displaystyle g(x)=f(Ax+b)}
7459:{\displaystyle h(x)=g(f(x))}
7280:{\displaystyle h(x)=g(f(x))}
5456:Lebesgue measurable function
3882:monotonically non-decreasing
3865:(an example is shown in the
3755:monotonically non-decreasing
3598:monotonically non-decreasing
2658:if and only if for all real
2086:{\displaystyle \mathbb {R} }
1799:{\displaystyle x_{1}=x_{2}.}
7:
13882:Hermite–Hadamard inequality
13633:Encyclopedia of Mathematics
13615:Encyclopedia of Mathematics
12897:Hermite–Hadamard inequality
12854:
12514:and convex on the interval
12507:{\displaystyle (0,\infty )}
12351:{\displaystyle (0,\infty )}
12271:which is greater than 0 if
12074:{\displaystyle g(x)=\log x}
11844:{\displaystyle 0<x<1}
11178:
10712:{\displaystyle f''(x)>0}
8513:{\displaystyle f''(x)>0}
7785:
7287:is convex. For example, if
7083:
6783:
6637:are all convex, then so is
6536:Nonnegative weighted sums:
3886:continuously differentiable
2915:A strictly convex function
2683:{\displaystyle 0<t<1}
2608:with the strict inequality
2199:{\displaystyle \pm \infty }
1195:{\displaystyle 0<t<1}
10:
14122:
13255:Dimitri Bertsekas (2003).
12669:positive-definite matrices
12326:is convex on the interval
12123:{\displaystyle h(x)=x^{2}}
11590:{\displaystyle f(x)=e^{x}}
11468:It is not strictly convex.
11319:{\displaystyle f(x)=x^{4}}
11224:{\displaystyle f(x)=x^{2}}
10821:Uniformly convex functions
9161:{\displaystyle \|\cdot \|}
8337:{\displaystyle a+b\leq 1.}
8170:{\displaystyle f(0)\leq 0}
6444:is concave if and only if
6405:{\displaystyle a,x,y>0}
6266:of two positive variables
5716:A differentiable function
4513:for positive real numbers
4413:{\displaystyle f(0)\leq 0}
4064:{\displaystyle f(x)=x^{4}}
3751:left and right derivatives
14067:
14034:
13989:
13920:
13846:
13770:
13712:
13686:
13521:Lauritzen, Niels (2013).
13512:, Rutickii Ya.B. (1961).
11184:Functions of one variable
10038:so the condition becomes
8348:Strongly convex functions
7871:is a convex set and that
7214:are convex functions and
6827:is convex. The domain of
6714:Elementwise maximum: let
6513:is convex if and only if
6026:Consequently, the set of
5509:{\displaystyle f(x,y)=xy}
3785:points, the set on which
3547:is convex if and only if
3178:Functions of one variable
2601:{\displaystyle \,\leq \,}
2101:extended real number line
14068:Applications and related
13872:Fenchel-Young inequality
12954:
12086:monotonically increasing
11999:monotonically increasing
11988:{\displaystyle x\leq 0.}
11748:with domain defined by
11431:{\displaystyle f(x)=|x|}
9817:{\displaystyle \succeq }
8279:{\displaystyle x,y\in X}
8142:is convex and satisfies
7336:{\displaystyle e^{f(x)}}
6487:is any real number then
6253:mathematical expectation
4610:{\displaystyle x_{2}=0,}
3121:is often referred to as
2572:{\displaystyle +\infty }
2549:{\displaystyle -\infty }
2504:are undefined). The sum
380:, can be used to deduce
13828:Legendre transformation
13752:Legendre transformation
13566:Thomson, Brian (1994).
13523:Undergraduate Convexity
13316:Yurii Nesterov (2004).
13014:Stewart, James (2015).
12161:{\displaystyle k(x)=-x}
12001:but not convex include
11958:{\displaystyle x\geq 0}
11542:{\displaystyle p\geq 1}
10031:{\displaystyle f''(x),}
9580:{\displaystyle m\to 0,}
8680:{\displaystyle (x_{n})}
5563:{\displaystyle f:X\to }
2625:{\displaystyle \,<.}
2099:that are valued in the
508:
208:{\displaystyle f(x)=cx}
14075:Convexity in economics
14009:(lower) ideally convex
13867:Fenchel–Moreau theorem
13857:Carathéodory's theorem
13575:Zălinescu, C. (2002).
13445:10.1093/qmath/12.1.283
12917:Kachurovskii's theorem
12818:
12757:
12692:
12661:
12604:
12578:
12543:
12508:
12476:
12441:
12387:
12352:
12320:
12291:
12290:{\displaystyle x>0}
12265:
12209:
12162:
12124:
12075:
12034:
11989:
11959:
11933:
11916:has second derivative
11910:
11880:
11879:{\displaystyle (0,1),}
11845:
11813:
11742:
11719:
11703:logarithmically convex
11695:
11644:
11591:
11543:
11517:
11462:
11432:
11376:
11320:
11271:
11225:
11169:
11146:
11092:
11072:
10926:
10925:{\displaystyle t\in ,}
10885:
10859:
10839:
10773:For every real number
10739:
10738:{\displaystyle x\in X}
10713:
10673:
10653:
10630:
10562:
10539:
10420:
10264:
10180:
10138:
10098:
10072:
10032:
9995:
9956:
9933:
9913:
9874:positive semi-definite
9866:
9818:
9794:
9764:
9744:
9724:
9676:
9656:
9633:
9632:{\displaystyle m>0}
9607:
9581:
9550:
9389:
9388:{\displaystyle t\in ,}
9348:
9322:
9297:
9162:
9132:
9100:
9004:
8893:
8867:
8866:{\displaystyle m>0}
8841:
8818:
8817:{\displaystyle f''(x)}
8784:
8737:
8681:
8648:
8625:
8602:
8556:
8534:
8514:
8474:
8453:
8430:
8396:convex if and only if
8390:
8367:
8338:
8306:
8280:
8248:
8171:
8136:
8102:
8079:
7992:
7928:
7906:
7865:
7845:
7822:
7756:
7724:
7723:{\displaystyle f(x,y)}
7685:
7640:
7578:
7525:
7489:is convex with domain
7483:
7460:
7407:
7387:
7364:
7337:
7307:is convex, then so is
7301:
7281:
7228:
7208:
7188:
7160:
7140:
7120:
7054:
7034:
7033:{\displaystyle f(x,y)}
6994:
6899:
6850:
6821:
6754:
6700:
6631:
6585:
6527:
6507:
6481:
6458:
6438:
6406:
6374:for all positive real
6368:
6303:
6280:
6245:
6225:
6154:
6131:
6111:
6070:
6044:
6009:
5981:
5934:
5884:
5867:For a convex function
5843:
5817:
5730:
5707:
5625:
5564:
5510:
5448:
5332:
5286:
5266:
5241:
5176:
4906:
4842:
4646:
4611:
4575:
4547:
4527:
4507:
4434:
4414:
4379:
4354:
4334:
4309:
4289:
4264:
4244:
4219:
4199:
4171:
4144:
4115:
4065:
4008:
3988:
3968:
3859:
3839:
3819:
3799:
3771:
3743:
3724:
3697:
3674:
3651:
3624:
3623:{\displaystyle x_{1},}
3590:
3541:
3521:
3494:
3467:
3447:
3394:
3374:
3325:
3197:
3155:
3103:
3083:
3046:
3023:
2969:
2949:
2929:
2907:
2770:
2730:
2684:
2646:
2626:
2602:
2573:
2550:
2527:
2498:
2463:
2437:
2357:
2307:
2260:
2240:
2220:
2200:
2177:
2163:where such a function
2157:
2087:
2061:
1999:
1937:
1800:
1757:
1734:
1708:
1646:
1584:
1561:
1541:
1480:
1419:
1282:
1242:
1196:
1161:
1141:
1124:-axis of the graph of
1118:
1098:
1078:
1051:
1024:
1004:
984:
964:
944:
924:
904:
884:
861:
841:
780:
719:
582:
536:
494:
471:
429:
412:
362:calculus of variations
339:
319:
318:{\displaystyle ce^{x}}
286:
266:
265:{\displaystyle cx^{2}}
229:
209:
149:
125:
78:
70:
44:
25:
20:Convex function on an
14101:Generalized convexity
13997:Convex series related
13897:Shapley–Folkman lemma
13370:C. Zalinescu (2002).
13072:Mathematical Analysis
12937:Pseudoconvex function
12927:Karamata's inequality
12819:
12758:
12693:
12676:linear transformation
12662:
12605:
12579:
12544:
12509:
12477:
12442:
12388:
12353:
12321:
12292:
12266:
12210:
12163:
12125:
12076:
12035:
11990:
11960:
11934:
11911:
11909:{\displaystyle x^{3}}
11881:
11846:
11814:
11743:
11720:
11696:
11645:
11592:
11544:
11518:
11463:
11433:
11377:
11321:
11272:
11226:
11170:
11168:{\displaystyle \phi }
11147:
11093:
11091:{\displaystyle \phi }
11073:
10927:
10886:
10860:
10840:
10838:{\displaystyle \phi }
10747:extreme value theorem
10740:
10714:
10674:
10654:
10631:
10563:
10540:
10421:
10265:
10181:
10139:
10099:
10073:
10033:
9996:
9957:
9934:
9914:
9867:
9819:
9804:, and the inequality
9795:
9765:
9750:in the domain, where
9745:
9725:
9677:
9657:
9639:(see example below).
9634:
9608:
9582:
9551:
9390:
9349:
9323:
9298:
9168:is the corresponding
9163:
9133:
9101:
9005:
8894:
8868:
8842:
8819:
8785:
8738:
8682:
8649:
8626:
8603:
8557:
8535:
8515:
8475:
8454:
8431:
8391:
8368:
8339:
8307:
8281:
8249:
8172:
8137:
8103:
8080:
7993:
7929:
7907:
7866:
7846:
7823:
7757:
7755:{\displaystyle (x,y)}
7725:
7686:
7641:
7579:
7526:
7484:
7461:
7408:
7388:
7365:
7363:{\displaystyle e^{x}}
7338:
7302:
7282:
7229:
7209:
7189:
7161:
7141:
7121:
7055:
7035:
6995:
6900:
6851:
6822:
6755:
6701:
6632:
6586:
6528:
6508:
6482:
6459:
6439:
6407:
6369:
6304:
6281:
6246:
6226:
6155:
6132:
6112:
6071:
6045:
6030:of a convex function
6010:
5982:
5935:
5885:
5862:positive semidefinite
5844:
5818:
5731:
5708:
5626:
5572:extended real numbers
5565:
5511:
5449:
5333:
5287:
5267:
5242:
5177:
4907:
4843:
4647:
4612:
4576:
4548:
4528:
4508:
4435:
4415:
4380:
4355:
4335:
4315:while its derivative
4310:
4290:
4265:
4245:
4220:
4200:
4172:
4170:{\displaystyle x^{4}}
4145:
4116:
4066:
4009:
3989:
3969:
3860:
3840:
3820:
3800:
3772:
3744:
3725:
3698:
3675:
3652:
3650:{\displaystyle x_{2}}
3625:
3591:
3542:
3522:
3520:{\displaystyle x_{2}}
3495:
3493:{\displaystyle x_{1}}
3468:
3448:
3395:
3375:
3326:
3203:is a function of one
3198:
3156:
3154:{\displaystyle \cup }
3133:is often referred as
3104:
3084:
3047:
3024:
2970:
2950:
2930:
2908:
2771:
2731:
2685:
2647:
2627:
2603:
2574:
2551:
2528:
2499:
2464:
2438:
2358:
2308:
2261:
2241:
2221:
2201:
2178:
2158:
2088:
2062:
2000:
1938:
1801:
1758:
1735:
1709:
1647:
1585:
1562:
1542:
1481:
1420:
1283:
1243:
1197:
1162:
1142:
1119:
1099:
1079:
1077:{\displaystyle x_{2}}
1052:
1050:{\displaystyle x_{1}}
1025:
1005:
985:
965:
945:
925:
905:
885:
862:
842:
781:
720:
583:
537:
495:
472:
430:
410:
356:has no more than one
340:
320:
287:
267:
230:
210:
150:
148:{\displaystyle \cap }
126:
124:{\displaystyle \cup }
99:graph of the function
77:Convex vs. Not convex
76:
50:
34:
19:
13887:Krein–Milman theorem
13680:variational analysis
13482:. Athena Scientific.
13406:. Springer. p.
13374:. World Scientific.
12949:of a convex function
12942:Quasiconvex function
12837:positive homogeneity
12774:
12702:
12682:
12633:
12603:{\displaystyle x=0.}
12588:
12553:
12518:
12486:
12451:
12400:
12362:
12330:
12319:{\displaystyle f(x)}
12301:
12275:
12219:
12175:
12134:
12092:
12044:
12005:
11973:
11943:
11920:
11893:
11855:
11823:
11752:
11732:
11709:
11654:
11601:
11559:
11554:exponential function
11527:
11475:
11461:{\displaystyle x=0.}
11446:
11397:
11330:
11288:
11235:
11193:
11159:
11102:
11082:
10936:
10895:
10869:
10849:
10829:
10723:
10683:
10663:
10659:on a compact domain
10643:
10576:
10552:
10430:
10274:
10194:
10151:
10108:
10082:
10042:
10005:
9966:
9943:
9923:
9884:
9828:
9808:
9774:
9770:is the identity and
9754:
9734:
9686:
9666:
9646:
9617:
9606:{\displaystyle m=0.}
9591:
9562:
9398:
9358:
9332:
9309:
9183:
9146:
9110:
9014:
9010:or, more generally,
8903:
8877:
8851:
8831:
8794:
8747:
8691:
8658:
8638:
8612:
8566:
8546:
8524:
8484:
8464:
8440:
8400:
8380:
8357:
8316:
8290:
8258:
8181:
8146:
8112:
8092:
8002:
7938:
7918:
7875:
7855:
7832:
7766:
7734:
7699:
7650:
7588:
7535:
7493:
7473:
7417:
7397:
7377:
7347:
7311:
7291:
7238:
7218:
7198:
7178:
7166:is not a convex set.
7150:
7130:
7064:
7044:
7009:
6909:
6863:
6849:{\displaystyle g(x)}
6831:
6764:
6718:
6641:
6595:
6543:
6517:
6491:
6471:
6448:
6425:
6378:
6313:
6290:
6270:
6264:homogeneous function
6235:
6164:
6141:
6121:
6101:
6054:
6034:
6019:quasiconvex function
5991:
5944:
5897:
5871:
5827:
5740:
5720:
5638:
5577:
5527:
5476:
5341:
5296:
5276:
5256:
5186:
4916:
4852:
4655:
4621:
4585:
4565:
4537:
4517:
4452:
4424:
4389:
4369:
4344:
4319:
4299:
4274:
4254:
4229:
4209:
4184:
4177:is strictly convex.
4154:
4143:{\displaystyle x=0,}
4125:
4121:, which is zero for
4075:
4033:
3998:
3978:
3899:
3849:
3829:
3809:
3789:
3761:
3733:
3711:
3687:
3664:
3634:
3604:
3551:
3531:
3504:
3477:
3457:
3408:
3384:
3335:
3211:
3187:
3145:
3093:
3070:
3036:
2979:
2959:
2939:
2919:
2780:
2740:
2694:
2662:
2636:
2632:Explicitly, the map
2612:
2590:
2560:
2537:
2508:
2473:
2447:
2367:
2317:
2270:
2250:
2230:
2210:
2187:
2167:
2106:
2075:
2009:
1947:
1810:
1767:
1744:
1718:
1656:
1594:
1571:
1551:
1490:
1429:
1292:
1252:
1206:
1174:
1151:
1128:
1108:
1088:
1061:
1034:
1014:
994:
974:
954:
934:
914:
894:
871:
851:
790:
729:
592:
546:
514:
484:
447:
419:
329:
299:
294:exponential function
276:
246:
219:
181:
139:
115:
87:real-valued function
13877:Jensen's inequality
13747:Lagrange multiplier
13737:Convex optimization
13732:Convex metric space
13549:. Wiley & Sons.
13510:Krasnosel'skii M.A.
13457:Cohen, J.E., 1981.
13132:Convex Optimization
13115:Extract of page 160
13058:Extract of page 227
12907:Jensen's inequality
12892:Hahn–Banach theorem
12882:Convex optimization
12833:triangle inequality
11440:triangle inequality
10884:{\displaystyle x,y}
10097:{\displaystyle m=0}
9545:
9347:{\displaystyle x,y}
9292:
8999:
8892:{\displaystyle x,y}
8480:strictly convex if
8305:{\displaystyle a,b}
8108:be a vector space.
6506:{\displaystyle r+f}
6257:Jensen's inequality
6095:Jensen's inequality
5858:partial derivatives
5842:{\displaystyle x,y}
4333:{\displaystyle f''}
4198:{\displaystyle f''}
3781:at all but at most
3163:Jensen's inequality
2955:is above the curve
2363:respectively, then
2183:is allowed to take
1733:{\displaystyle t=0}
390:Hölder's inequality
378:Jensen's inequality
14106:Types of functions
14005:(cs, bcs)-complete
13976:Algebraic interior
13694:Convex combination
13555:Rockafellar, R. T.
13503:Lemaréchal, Claude
13476:Bertsekas, Dimitri
12887:Geodesic convexity
12848:nonnegative matrix
12814:
12766:Every real-valued
12753:
12688:
12674:Every real-valued
12657:
12600:
12574:
12539:
12504:
12472:
12437:
12435:
12383:
12348:
12316:
12287:
12261:
12259:
12205:
12203:
12158:
12120:
12071:
12030:
11985:
11955:
11932:{\displaystyle 6x}
11929:
11906:
11876:
11841:
11809:
11738:
11715:
11691:
11640:
11587:
11539:
11513:
11458:
11428:
11372:
11316:
11267:
11221:
11165:
11142:
11130:
11088:
11068:
10922:
10891:in the domain and
10881:
10855:
10835:
10735:
10709:
10669:
10649:
10626:
10558:
10535:
10416:
10260:
10176:
10134:
10094:
10068:
10028:
9991:
9955:{\displaystyle x.}
9952:
9929:
9909:
9862:
9814:
9790:
9760:
9740:
9720:
9672:
9652:
9629:
9603:
9577:
9546:
9531:
9385:
9354:in the domain and
9344:
9321:{\displaystyle m,}
9318:
9293:
9278:
9158:
9128:
9096:
9000:
8985:
8889:
8863:
8837:
8814:
8780:
8733:
8731:
8677:
8644:
8624:{\displaystyle x.}
8621:
8598:
8552:
8530:
8510:
8470:
8452:{\displaystyle x.}
8449:
8426:
8386:
8363:
8334:
8302:
8276:
8244:
8167:
8132:
8098:
8075:
8038:
7988:
7982:
7924:
7902:
7861:
7844:{\displaystyle x,}
7841:
7818:
7752:
7720:
7681:
7636:
7574:
7521:
7479:
7456:
7403:
7383:
7360:
7333:
7297:
7277:
7224:
7204:
7184:
7156:
7136:
7116:
7050:
7030:
6990:
6895:
6846:
6817:
6750:
6696:
6627:
6581:
6523:
6503:
6477:
6454:
6437:{\displaystyle -f}
6434:
6402:
6364:
6302:{\displaystyle y,}
6299:
6276:
6241:
6221:
6153:{\displaystyle f,}
6150:
6127:
6107:
6066:
6040:
6005:
5977:
5930:
5883:{\displaystyle f,}
5880:
5839:
5813:
5726:
5703:
5621:
5560:
5506:
5444:
5328:
5282:
5262:
5237:
5172:
5170:
4912:, it follows that
4902:
4838:
4836:
4642:
4607:
4571:
4559:
4543:
4523:
4503:
4430:
4410:
4375:
4350:
4330:
4305:
4288:{\displaystyle f'}
4285:
4260:
4243:{\displaystyle f'}
4240:
4215:
4195:
4167:
4140:
4111:
4061:
4004:
3984:
3964:
3855:
3835:
3815:
3795:
3767:
3739:
3723:{\displaystyle C.}
3720:
3693:
3670:
3660:A convex function
3647:
3620:
3586:
3537:
3517:
3490:
3463:
3443:
3390:
3370:
3321:
3193:
3161:. As an example,
3151:
3113:Alternative naming
3099:
3082:{\displaystyle -f}
3079:
3042:
3019:
2965:
2945:
2925:
2903:
2766:
2726:
2680:
2642:
2622:
2598:
2569:
2546:
2523:
2494:
2459:
2433:
2353:
2303:
2256:
2236:
2216:
2196:
2173:
2153:
2083:
2057:
1995:
1933:
1796:
1756:{\displaystyle 1,}
1753:
1730:
1704:
1642:
1583:{\displaystyle f;}
1580:
1557:
1537:
1476:
1415:
1278:
1238:
1192:
1157:
1140:{\displaystyle f.}
1137:
1114:
1094:
1074:
1047:
1020:
1000:
980:
960:
940:
920:
900:
883:{\displaystyle t;}
880:
857:
837:
776:
715:
578:
532:
490:
467:
425:
413:
366:probability theory
335:
315:
282:
262:
241:quadratic function
225:
205:
145:
121:
79:
71:
45:
26:
14083:
14082:
13543:Luenberger, David
13537:. Addison-Wesley.
13531:Luenberger, David
13497:. Academic Press.
13487:Borwein, Jonathan
13417:978-1-4419-9467-7
13145:978-0-521-83378-3
13109:978-1-108-48053-6
13082:978-5-03-000500-3
13052:978-0-486-13887-9
12969:"Lecture Notes 2"
12923:of the derivative
12912:K-convex function
12691:{\displaystyle f}
12667:on the domain of
12434:
12258:
12202:
12028:
11969:on the set where
11741:{\displaystyle f}
11718:{\displaystyle f}
11129:
10858:{\displaystyle f}
10672:{\displaystyle X}
10652:{\displaystyle f}
10608:
10561:{\displaystyle f}
10355:
9932:{\displaystyle m}
9763:{\displaystyle I}
9743:{\displaystyle x}
9675:{\displaystyle m}
9655:{\displaystyle f}
9496:
9328:is that, for all
9264:
8840:{\displaystyle f}
8730:
8647:{\displaystyle f}
8634:For example, let
8555:{\displaystyle f}
8533:{\displaystyle x}
8473:{\displaystyle f}
8389:{\displaystyle f}
8366:{\displaystyle f}
8101:{\displaystyle X}
8037:
7981:
7927:{\displaystyle f}
7864:{\displaystyle C}
7695:Minimization: If
7482:{\displaystyle f}
7406:{\displaystyle g}
7386:{\displaystyle f}
7300:{\displaystyle f}
7227:{\displaystyle g}
7207:{\displaystyle g}
7187:{\displaystyle f}
7159:{\displaystyle C}
7139:{\displaystyle x}
7053:{\displaystyle x}
7003:Danskin's theorem
6526:{\displaystyle f}
6480:{\displaystyle r}
6457:{\displaystyle f}
6279:{\displaystyle x}
6130:{\displaystyle X}
6110:{\displaystyle f}
6050:is a convex set:
6043:{\displaystyle f}
6028:global minimisers
5729:{\displaystyle f}
5681:
5675:
5518:marginally linear
5439:
5382:
5285:{\displaystyle C}
5265:{\displaystyle f}
5117:
5078:
5045:
4996:
4574:{\displaystyle f}
4557:
4546:{\displaystyle b}
4526:{\displaystyle a}
4433:{\displaystyle f}
4378:{\displaystyle f}
4353:{\displaystyle X}
4308:{\displaystyle X}
4263:{\displaystyle X}
4218:{\displaystyle X}
4023:inflection points
4019:second derivative
4007:{\displaystyle y}
3987:{\displaystyle x}
3858:{\displaystyle C}
3838:{\displaystyle f}
3818:{\displaystyle C}
3798:{\displaystyle f}
3770:{\displaystyle f}
3742:{\displaystyle f}
3696:{\displaystyle C}
3673:{\displaystyle f}
3540:{\displaystyle f}
3466:{\displaystyle R}
3393:{\displaystyle R}
3319:
3196:{\displaystyle f}
3102:{\displaystyle f}
3045:{\displaystyle f}
2968:{\displaystyle f}
2948:{\displaystyle f}
2928:{\displaystyle f}
2645:{\displaystyle f}
2259:{\displaystyle 1}
2239:{\displaystyle 0}
2219:{\displaystyle t}
2176:{\displaystyle f}
1560:{\displaystyle f}
1160:{\displaystyle f}
1117:{\displaystyle x}
1097:{\displaystyle X}
1023:{\displaystyle f}
1003:{\displaystyle 0}
983:{\displaystyle 1}
963:{\displaystyle t}
943:{\displaystyle 1}
923:{\displaystyle 0}
903:{\displaystyle t}
867:as a function of
860:{\displaystyle f}
493:{\displaystyle f}
428:{\displaystyle X}
408:
338:{\displaystyle c}
285:{\displaystyle c}
228:{\displaystyle c}
168:second derivative
14113:
14001:(cs, lcs)-closed
13947:Effective domain
13902:Robinson–Ursescu
13778:Convex conjugate
13669:
13662:
13655:
13646:
13645:
13641:
13623:
13598:
13571:
13562:
13550:
13538:
13526:
13517:
13498:
13483:
13462:
13455:
13449:
13448:
13428:
13422:
13421:
13405:
13395:
13386:
13385:
13367:
13358:
13357:
13355:
13346:
13340:
13339:
13323:
13313:
13304:
13303:
13285:
13279:
13278:
13262:
13252:
13246:
13245:
13243:
13242:
13228:
13222:
13219:
13213:
13212:
13210:
13208:
13197:
13191:
13190:
13188:
13186:
13163:
13157:
13156:
13154:
13152:
13137:
13126:
13117:
13113:
13093:
13087:
13086:
13066:
13060:
13056:
13036:
13030:
13029:
13011:
13005:
13004:
12993:
12987:
12986:
12984:
12982:
12976:www.stat.cmu.edu
12973:
12965:
12872:Convex conjugate
12862:Concave function
12823:
12821:
12820:
12815:
12801:
12800:
12762:
12760:
12759:
12754:
12698:is linear, then
12697:
12695:
12694:
12689:
12666:
12664:
12663:
12658:
12609:
12607:
12606:
12601:
12583:
12581:
12580:
12575:
12548:
12546:
12545:
12540:
12513:
12511:
12510:
12505:
12481:
12479:
12478:
12473:
12446:
12444:
12443:
12438:
12436:
12433:
12432:
12420:
12392:
12390:
12389:
12384:
12357:
12355:
12354:
12349:
12325:
12323:
12322:
12317:
12296:
12294:
12293:
12288:
12270:
12268:
12267:
12262:
12260:
12257:
12256:
12244:
12229:
12214:
12212:
12211:
12206:
12204:
12195:
12167:
12165:
12164:
12159:
12129:
12127:
12126:
12121:
12119:
12118:
12080:
12078:
12077:
12072:
12039:
12037:
12036:
12031:
12029:
12024:
11994:
11992:
11991:
11986:
11964:
11962:
11961:
11956:
11938:
11936:
11935:
11930:
11915:
11913:
11912:
11907:
11905:
11904:
11885:
11883:
11882:
11877:
11850:
11848:
11847:
11842:
11818:
11816:
11815:
11810:
11747:
11745:
11744:
11739:
11724:
11722:
11721:
11716:
11700:
11698:
11697:
11692:
11690:
11689:
11649:
11647:
11646:
11641:
11633:
11632:
11611:
11596:
11594:
11593:
11588:
11586:
11585:
11548:
11546:
11545:
11540:
11522:
11520:
11519:
11514:
11512:
11511:
11506:
11497:
11467:
11465:
11464:
11459:
11437:
11435:
11434:
11429:
11427:
11419:
11385:
11381:
11379:
11378:
11373:
11365:
11364:
11340:
11325:
11323:
11322:
11317:
11315:
11314:
11280:
11276:
11274:
11273:
11268:
11245:
11230:
11228:
11227:
11222:
11220:
11219:
11174:
11172:
11171:
11166:
11151:
11149:
11148:
11143:
11141:
11140:
11131:
11122:
11097:
11095:
11094:
11089:
11077:
11075:
11074:
11069:
10931:
10929:
10928:
10923:
10890:
10888:
10887:
10882:
10864:
10862:
10861:
10856:
10845:, is a function
10844:
10842:
10841:
10836:
10744:
10742:
10741:
10736:
10718:
10716:
10715:
10710:
10693:
10678:
10676:
10675:
10670:
10658:
10656:
10655:
10650:
10635:
10633:
10632:
10627:
10625:
10624:
10609:
10601:
10567:
10565:
10564:
10559:
10544:
10542:
10541:
10536:
10519:
10518:
10464:
10463:
10454:
10453:
10425:
10423:
10422:
10417:
10388:
10387:
10378:
10377:
10356:
10348:
10328:
10327:
10269:
10267:
10266:
10261:
10188:Taylor's Theorem
10185:
10183:
10182:
10177:
10163:
10162:
10143:
10141:
10140:
10135:
10118:
10103:
10101:
10100:
10095:
10077:
10075:
10074:
10069:
10052:
10037:
10035:
10034:
10029:
10015:
10000:
9998:
9997:
9992:
9978:
9977:
9961:
9959:
9958:
9953:
9938:
9936:
9935:
9930:
9918:
9916:
9915:
9910:
9896:
9895:
9871:
9869:
9868:
9863:
9840:
9839:
9823:
9821:
9820:
9815:
9799:
9797:
9796:
9791:
9786:
9785:
9769:
9767:
9766:
9761:
9749:
9747:
9746:
9741:
9729:
9727:
9726:
9721:
9698:
9697:
9681:
9679:
9678:
9673:
9661:
9659:
9658:
9653:
9642:If the function
9638:
9636:
9635:
9630:
9612:
9610:
9609:
9604:
9586:
9584:
9583:
9578:
9555:
9553:
9552:
9547:
9544:
9539:
9497:
9489:
9394:
9392:
9391:
9386:
9353:
9351:
9350:
9345:
9327:
9325:
9324:
9319:
9302:
9300:
9299:
9294:
9291:
9286:
9265:
9257:
9237:
9236:
9167:
9165:
9164:
9159:
9137:
9135:
9134:
9129:
9105:
9103:
9102:
9097:
9095:
9094:
9009:
9007:
9006:
9001:
8998:
8993:
8951:
8950:
8898:
8896:
8895:
8890:
8872:
8870:
8869:
8864:
8846:
8844:
8843:
8838:
8823:
8821:
8820:
8815:
8804:
8789:
8787:
8786:
8781:
8770:
8769:
8757:
8742:
8740:
8739:
8734:
8732:
8723:
8714:
8713:
8701:
8686:
8684:
8683:
8678:
8673:
8672:
8653:
8651:
8650:
8645:
8630:
8628:
8627:
8622:
8607:
8605:
8604:
8599:
8576:
8561:
8559:
8558:
8553:
8539:
8537:
8536:
8531:
8519:
8517:
8516:
8511:
8494:
8479:
8477:
8476:
8471:
8458:
8456:
8455:
8450:
8435:
8433:
8432:
8427:
8410:
8395:
8393:
8392:
8387:
8372:
8370:
8369:
8364:
8343:
8341:
8340:
8335:
8311:
8309:
8308:
8303:
8285:
8283:
8282:
8277:
8253:
8251:
8250:
8245:
8176:
8174:
8173:
8168:
8141:
8139:
8138:
8133:
8131:
8107:
8105:
8104:
8099:
8084:
8082:
8081:
8076:
8074:
8070:
8039:
8030:
7997:
7995:
7994:
7989:
7987:
7983:
7974:
7933:
7931:
7930:
7925:
7911:
7909:
7908:
7903:
7870:
7868:
7867:
7862:
7850:
7848:
7847:
7842:
7827:
7825:
7824:
7819:
7799:
7798:
7761:
7759:
7758:
7753:
7729:
7727:
7726:
7721:
7690:
7688:
7687:
7682:
7677:
7676:
7671:
7662:
7661:
7645:
7643:
7642:
7637:
7635:
7634:
7629:
7614:
7613:
7602:
7583:
7581:
7580:
7575:
7530:
7528:
7527:
7522:
7520:
7519:
7514:
7505:
7504:
7488:
7486:
7485:
7480:
7465:
7463:
7462:
7457:
7412:
7410:
7409:
7404:
7392:
7390:
7389:
7384:
7369:
7367:
7366:
7361:
7359:
7358:
7342:
7340:
7339:
7334:
7332:
7331:
7306:
7304:
7303:
7298:
7286:
7284:
7283:
7278:
7233:
7231:
7230:
7225:
7213:
7211:
7210:
7205:
7193:
7191:
7190:
7185:
7165:
7163:
7162:
7157:
7145:
7143:
7142:
7137:
7125:
7123:
7122:
7117:
7097:
7096:
7059:
7057:
7056:
7051:
7039:
7037:
7036:
7031:
6999:
6997:
6996:
6991:
6986:
6982:
6972:
6971:
6944:
6943:
6904:
6902:
6901:
6896:
6894:
6893:
6875:
6874:
6855:
6853:
6852:
6847:
6826:
6824:
6823:
6818:
6807:
6806:
6797:
6796:
6759:
6757:
6756:
6751:
6749:
6748:
6733:
6732:
6705:
6703:
6702:
6697:
6692:
6691:
6682:
6681:
6663:
6662:
6653:
6652:
6636:
6634:
6633:
6628:
6626:
6625:
6607:
6606:
6590:
6588:
6587:
6582:
6574:
6573:
6555:
6554:
6532:
6530:
6529:
6524:
6512:
6510:
6509:
6504:
6486:
6484:
6483:
6478:
6463:
6461:
6460:
6455:
6443:
6441:
6440:
6435:
6411:
6409:
6408:
6403:
6373:
6371:
6370:
6365:
6308:
6306:
6305:
6300:
6285:
6283:
6282:
6277:
6250:
6248:
6247:
6242:
6230:
6228:
6227:
6222:
6159:
6157:
6156:
6151:
6136:
6134:
6133:
6128:
6116:
6114:
6113:
6108:
6075:
6073:
6072:
6067:
6061:
6049:
6047:
6046:
6041:
6014:
6012:
6011:
6006:
6004:
5986:
5984:
5983:
5978:
5939:
5937:
5936:
5931:
5889:
5887:
5886:
5881:
5848:
5846:
5845:
5840:
5822:
5820:
5819:
5814:
5794:
5793:
5735:
5733:
5732:
5727:
5713:is a convex set.
5712:
5710:
5709:
5704:
5679:
5673:
5672:
5630:
5628:
5627:
5622:
5605:
5569:
5567:
5566:
5561:
5515:
5513:
5512:
5507:
5453:
5451:
5450:
5445:
5440:
5435:
5431:
5430:
5409:
5408:
5392:
5387:
5383:
5378:
5377:
5376:
5364:
5363:
5353:
5337:
5335:
5334:
5329:
5321:
5320:
5308:
5307:
5291:
5289:
5288:
5283:
5271:
5269:
5268:
5263:
5246:
5244:
5243:
5238:
5181:
5179:
5178:
5173:
5171:
5140:
5118:
5116:
5102:
5079:
5077:
5063:
5055:
5051:
5047:
5046:
5044:
5030:
5002:
4998:
4997:
4995:
4981:
4911:
4909:
4908:
4903:
4898:
4897:
4873:
4872:
4847:
4845:
4844:
4839:
4837:
4827:
4826:
4802:
4765:
4764:
4740:
4709:
4708:
4680:
4679:
4651:
4649:
4648:
4643:
4616:
4614:
4613:
4608:
4597:
4596:
4580:
4578:
4577:
4572:
4552:
4550:
4549:
4544:
4532:
4530:
4529:
4524:
4512:
4510:
4509:
4504:
4439:
4437:
4436:
4431:
4419:
4417:
4416:
4411:
4384:
4382:
4381:
4376:
4359:
4357:
4356:
4351:
4339:
4337:
4336:
4331:
4329:
4314:
4312:
4311:
4306:
4294:
4292:
4291:
4286:
4284:
4269:
4267:
4266:
4261:
4249:
4247:
4246:
4241:
4239:
4224:
4222:
4221:
4216:
4204:
4202:
4201:
4196:
4194:
4176:
4174:
4173:
4168:
4166:
4165:
4149:
4147:
4146:
4141:
4120:
4118:
4117:
4112:
4110:
4109:
4085:
4070:
4068:
4067:
4062:
4060:
4059:
4014:in the interval.
4013:
4011:
4010:
4005:
3993:
3991:
3990:
3985:
3973:
3971:
3970:
3965:
3939:
3867:examples section
3864:
3862:
3861:
3856:
3844:
3842:
3841:
3836:
3825:is closed, then
3824:
3822:
3821:
3816:
3804:
3802:
3801:
3796:
3776:
3774:
3773:
3768:
3753:, and these are
3748:
3746:
3745:
3740:
3729:
3727:
3726:
3721:
3702:
3700:
3699:
3694:
3679:
3677:
3676:
3671:
3656:
3654:
3653:
3648:
3646:
3645:
3630:for every fixed
3629:
3627:
3626:
3621:
3616:
3615:
3595:
3593:
3592:
3587:
3582:
3581:
3569:
3568:
3546:
3544:
3543:
3538:
3526:
3524:
3523:
3518:
3516:
3515:
3499:
3497:
3496:
3491:
3489:
3488:
3472:
3470:
3469:
3464:
3452:
3450:
3449:
3444:
3436:
3435:
3423:
3422:
3399:
3397:
3396:
3391:
3379:
3377:
3376:
3371:
3366:
3365:
3353:
3352:
3330:
3328:
3327:
3322:
3320:
3318:
3317:
3316:
3304:
3303:
3293:
3289:
3288:
3267:
3266:
3250:
3242:
3241:
3229:
3228:
3202:
3200:
3199:
3194:
3160:
3158:
3157:
3152:
3108:
3106:
3105:
3100:
3088:
3086:
3085:
3080:
3063:strictly concave
3051:
3049:
3048:
3043:
3028:
3026:
3025:
3020:
3012:
3011:
2974:
2972:
2971:
2966:
2954:
2952:
2951:
2946:
2934:
2932:
2931:
2926:
2912:
2910:
2909:
2904:
2902:
2898:
2897:
2863:
2859:
2858:
2836:
2832:
2831:
2830:
2803:
2802:
2775:
2773:
2772:
2767:
2765:
2764:
2752:
2751:
2735:
2733:
2732:
2727:
2719:
2718:
2706:
2705:
2689:
2687:
2686:
2681:
2651:
2649:
2648:
2643:
2631:
2629:
2628:
2623:
2607:
2605:
2604:
2599:
2584:strict convexity
2578:
2576:
2575:
2570:
2555:
2553:
2552:
2547:
2532:
2530:
2529:
2524:
2503:
2501:
2500:
2495:
2468:
2466:
2465:
2460:
2442:
2440:
2439:
2434:
2432:
2428:
2427:
2393:
2389:
2388:
2362:
2360:
2359:
2354:
2340:
2336:
2335:
2312:
2310:
2309:
2304:
2293:
2289:
2288:
2265:
2263:
2262:
2257:
2245:
2243:
2242:
2237:
2225:
2223:
2222:
2217:
2205:
2203:
2202:
2197:
2182:
2180:
2179:
2174:
2162:
2160:
2159:
2154:
2134:
2096:convex functions
2092:
2090:
2089:
2084:
2082:
2066:
2064:
2063:
2058:
2056:
2052:
2051:
2032:
2028:
2027:
2004:
2002:
2001:
1996:
1994:
1990:
1989:
1970:
1966:
1965:
1942:
1940:
1939:
1934:
1932:
1928:
1927:
1893:
1889:
1888:
1866:
1862:
1861:
1860:
1833:
1832:
1805:
1803:
1802:
1797:
1792:
1791:
1779:
1778:
1762:
1760:
1759:
1754:
1739:
1737:
1736:
1731:
1713:
1711:
1710:
1705:
1703:
1699:
1698:
1679:
1675:
1674:
1651:
1649:
1648:
1643:
1641:
1637:
1636:
1617:
1613:
1612:
1589:
1587:
1586:
1581:
1566:
1564:
1563:
1558:
1546:
1544:
1543:
1538:
1536:
1532:
1531:
1527:
1526:
1507:
1506:
1485:
1483:
1482:
1477:
1475:
1471:
1470:
1466:
1465:
1446:
1445:
1424:
1422:
1421:
1416:
1414:
1410:
1409:
1375:
1371:
1370:
1348:
1344:
1343:
1342:
1315:
1314:
1287:
1285:
1284:
1279:
1277:
1276:
1264:
1263:
1247:
1245:
1244:
1239:
1231:
1230:
1218:
1217:
1201:
1199:
1198:
1193:
1166:
1164:
1163:
1158:
1146:
1144:
1143:
1138:
1123:
1121:
1120:
1115:
1103:
1101:
1100:
1095:
1083:
1081:
1080:
1075:
1073:
1072:
1056:
1054:
1053:
1048:
1046:
1045:
1029:
1027:
1026:
1021:
1009:
1007:
1006:
1001:
989:
987:
986:
981:
969:
967:
966:
961:
949:
947:
946:
941:
929:
927:
926:
921:
909:
907:
906:
901:
889:
887:
886:
881:
866:
864:
863:
858:
847:in the graph of
846:
844:
843:
838:
836:
832:
831:
827:
826:
807:
806:
785:
783:
782:
777:
775:
771:
770:
766:
765:
746:
745:
724:
722:
721:
716:
714:
710:
709:
675:
671:
670:
648:
644:
643:
642:
615:
614:
587:
585:
584:
579:
571:
570:
558:
557:
541:
539:
538:
533:
499:
497:
496:
491:
476:
474:
473:
468:
466:
434:
432:
431:
426:
409:
344:
342:
341:
336:
324:
322:
321:
316:
314:
313:
291:
289:
288:
283:
271:
269:
268:
263:
261:
260:
234:
232:
231:
226:
214:
212:
211:
206:
154:
152:
151:
146:
133:concave function
130:
128:
127:
122:
68:
55:convex function
39:(in green) is a
14121:
14120:
14116:
14115:
14114:
14112:
14111:
14110:
14096:Convex analysis
14086:
14085:
14084:
14079:
14063:
14030:
13985:
13916:
13842:
13833:Semi-continuity
13818:Convex function
13799:Logarithmically
13766:
13727:Convex geometry
13708:
13699:Convex function
13682:
13676:Convex analysis
13673:
13626:
13608:
13605:
13587:
13559:Convex analysis
13471:
13466:
13465:
13456:
13452:
13429:
13425:
13418:
13396:
13389:
13382:
13368:
13361:
13353:
13347:
13343:
13336:
13314:
13307:
13300:
13286:
13282:
13275:
13253:
13249:
13240:
13238:
13230:
13229:
13225:
13220:
13216:
13206:
13204:
13199:
13198:
13194:
13184:
13182:
13180:
13164:
13160:
13150:
13148:
13146:
13135:
13127:
13120:
13110:
13094:
13090:
13083:
13067:
13063:
13053:
13037:
13033:
13026:
13012:
13008:
12995:
12994:
12990:
12980:
12978:
12971:
12967:
12966:
12962:
12957:
12952:
12867:Convex analysis
12857:
12844:spectral radius
12796:
12792:
12775:
12772:
12771:
12768:affine function
12703:
12700:
12699:
12683:
12680:
12679:
12634:
12631:
12630:
12620:
12589:
12586:
12585:
12554:
12551:
12550:
12519:
12516:
12515:
12487:
12484:
12483:
12452:
12449:
12448:
12428:
12424:
12418:
12401:
12398:
12397:
12363:
12360:
12359:
12331:
12328:
12327:
12302:
12299:
12298:
12276:
12273:
12272:
12252:
12248:
12242:
12222:
12220:
12217:
12216:
12193:
12176:
12173:
12172:
12135:
12132:
12131:
12114:
12110:
12093:
12090:
12089:
12045:
12042:
12041:
12023:
12006:
12003:
12002:
11974:
11971:
11970:
11944:
11941:
11940:
11921:
11918:
11917:
11900:
11896:
11894:
11891:
11890:
11856:
11853:
11852:
11824:
11821:
11820:
11753:
11750:
11749:
11733:
11730:
11729:
11710:
11707:
11706:
11676:
11672:
11655:
11652:
11651:
11628:
11624:
11604:
11602:
11599:
11598:
11581:
11577:
11560:
11557:
11556:
11528:
11525:
11524:
11507:
11502:
11501:
11493:
11476:
11473:
11472:
11447:
11444:
11443:
11423:
11415:
11398:
11395:
11394:
11383:
11360:
11356:
11333:
11331:
11328:
11327:
11310:
11306:
11289:
11286:
11285:
11278:
11238:
11236:
11233:
11232:
11215:
11211:
11194:
11191:
11190:
11186:
11181:
11160:
11157:
11156:
11136:
11132:
11120:
11103:
11100:
11099:
11083:
11080:
11079:
10937:
10934:
10933:
10896:
10893:
10892:
10870:
10867:
10866:
10850:
10847:
10846:
10830:
10827:
10826:
10823:
10759:
10724:
10721:
10720:
10686:
10684:
10681:
10680:
10679:that satisfies
10664:
10661:
10660:
10644:
10641:
10640:
10620:
10616:
10600:
10577:
10574:
10573:
10553:
10550:
10549:
10514:
10510:
10459:
10455:
10449:
10445:
10431:
10428:
10427:
10383:
10379:
10373:
10369:
10347:
10323:
10319:
10275:
10272:
10271:
10195:
10192:
10191:
10158:
10154:
10152:
10149:
10148:
10111:
10109:
10106:
10105:
10083:
10080:
10079:
10045:
10043:
10040:
10039:
10008:
10006:
10003:
10002:
9973:
9969:
9967:
9964:
9963:
9944:
9941:
9940:
9924:
9921:
9920:
9891:
9887:
9885:
9882:
9881:
9835:
9831:
9829:
9826:
9825:
9809:
9806:
9805:
9781:
9777:
9775:
9772:
9771:
9755:
9752:
9751:
9735:
9732:
9731:
9693:
9689:
9687:
9684:
9683:
9682:if and only if
9667:
9664:
9663:
9647:
9644:
9643:
9618:
9615:
9614:
9592:
9589:
9588:
9563:
9560:
9559:
9540:
9535:
9488:
9399:
9396:
9395:
9359:
9356:
9355:
9333:
9330:
9329:
9310:
9307:
9306:
9287:
9282:
9256:
9232:
9228:
9184:
9181:
9180:
9147:
9144:
9143:
9111:
9108:
9107:
9090:
9086:
9015:
9012:
9011:
8994:
8989:
8946:
8942:
8904:
8901:
8900:
8878:
8875:
8874:
8852:
8849:
8848:
8832:
8829:
8828:
8797:
8795:
8792:
8791:
8765:
8761:
8750:
8748:
8745:
8744:
8721:
8709:
8705:
8694:
8692:
8689:
8688:
8668:
8664:
8659:
8656:
8655:
8639:
8636:
8635:
8613:
8610:
8609:
8569:
8567:
8564:
8563:
8547:
8544:
8543:
8525:
8522:
8521:
8487:
8485:
8482:
8481:
8465:
8462:
8461:
8441:
8438:
8437:
8403:
8401:
8398:
8397:
8381:
8378:
8377:
8358:
8355:
8354:
8350:
8317:
8314:
8313:
8291:
8288:
8287:
8259:
8256:
8255:
8182:
8179:
8178:
8177:if and only if
8147:
8144:
8143:
8127:
8113:
8110:
8109:
8093:
8090:
8089:
8028:
8009:
8005:
8003:
8000:
7999:
7972:
7968:
7939:
7936:
7935:
7919:
7916:
7915:
7876:
7873:
7872:
7856:
7853:
7852:
7833:
7830:
7829:
7788:
7784:
7767:
7764:
7763:
7735:
7732:
7731:
7700:
7697:
7696:
7672:
7667:
7666:
7657:
7653:
7651:
7648:
7647:
7630:
7625:
7624:
7603:
7598:
7597:
7589:
7586:
7585:
7536:
7533:
7532:
7515:
7510:
7509:
7500:
7496:
7494:
7491:
7490:
7474:
7471:
7470:
7418:
7415:
7414:
7398:
7395:
7394:
7393:is concave and
7378:
7375:
7374:
7354:
7350:
7348:
7345:
7344:
7318:
7314:
7312:
7309:
7308:
7292:
7289:
7288:
7239:
7236:
7235:
7219:
7216:
7215:
7199:
7196:
7195:
7179:
7176:
7175:
7151:
7148:
7147:
7131:
7128:
7127:
7086:
7082:
7065:
7062:
7061:
7045:
7042:
7041:
7010:
7007:
7006:
6967:
6963:
6939:
6935:
6934:
6930:
6910:
6907:
6906:
6889:
6885:
6870:
6866:
6864:
6861:
6860:
6832:
6829:
6828:
6802:
6798:
6786:
6782:
6765:
6762:
6761:
6738:
6734:
6728:
6724:
6719:
6716:
6715:
6687:
6683:
6677:
6673:
6658:
6654:
6648:
6644:
6642:
6639:
6638:
6621:
6617:
6602:
6598:
6596:
6593:
6592:
6569:
6565:
6550:
6546:
6544:
6541:
6540:
6518:
6515:
6514:
6492:
6489:
6488:
6472:
6469:
6468:
6449:
6446:
6445:
6426:
6423:
6422:
6419:
6379:
6376:
6375:
6314:
6311:
6310:
6291:
6288:
6287:
6271:
6268:
6267:
6236:
6233:
6232:
6165:
6162:
6161:
6142:
6139:
6138:
6122:
6119:
6118:
6102:
6099:
6098:
6057:
6055:
6052:
6051:
6035:
6032:
6031:
6000:
5992:
5989:
5988:
5945:
5942:
5941:
5898:
5895:
5894:
5872:
5869:
5868:
5828:
5825:
5824:
5789:
5785:
5741:
5738:
5737:
5721:
5718:
5717:
5668:
5639:
5636:
5635:
5601:
5578:
5575:
5574:
5528:
5525:
5524:
5477:
5474:
5473:
5469:
5426:
5422:
5404:
5400:
5393:
5391:
5372:
5368:
5359:
5355:
5354:
5352:
5348:
5342:
5339:
5338:
5316:
5312:
5303:
5299:
5297:
5294:
5293:
5277:
5274:
5273:
5257:
5254:
5253:
5249:
5187:
5184:
5183:
5169:
5168:
5138:
5137:
5106:
5101:
5067:
5062:
5053:
5052:
5034:
5029:
5013:
5009:
4985:
4980:
4964:
4960:
4950:
4919:
4917:
4914:
4913:
4893:
4889:
4868:
4864:
4853:
4850:
4849:
4835:
4834:
4822:
4818:
4800:
4799:
4760:
4756:
4738:
4737:
4704:
4700:
4684:
4675:
4671:
4658:
4656:
4653:
4652:
4622:
4619:
4618:
4592:
4588:
4586:
4583:
4582:
4566:
4563:
4562:
4538:
4535:
4534:
4518:
4515:
4514:
4453:
4450:
4449:
4425:
4422:
4421:
4390:
4387:
4386:
4370:
4367:
4366:
4345:
4342:
4341:
4322:
4320:
4317:
4316:
4300:
4297:
4296:
4277:
4275:
4272:
4271:
4255:
4252:
4251:
4232:
4230:
4227:
4226:
4210:
4207:
4206:
4187:
4185:
4182:
4181:
4161:
4157:
4155:
4152:
4151:
4126:
4123:
4122:
4105:
4101:
4078:
4076:
4073:
4072:
4055:
4051:
4034:
4031:
4030:
3999:
3996:
3995:
3979:
3976:
3975:
3932:
3900:
3897:
3896:
3850:
3847:
3846:
3830:
3827:
3826:
3810:
3807:
3806:
3790:
3787:
3786:
3762:
3759:
3758:
3734:
3731:
3730:
3712:
3709:
3708:
3688:
3685:
3684:
3665:
3662:
3661:
3641:
3637:
3635:
3632:
3631:
3611:
3607:
3605:
3602:
3601:
3577:
3573:
3564:
3560:
3552:
3549:
3548:
3532:
3529:
3528:
3511:
3507:
3505:
3502:
3501:
3484:
3480:
3478:
3475:
3474:
3458:
3455:
3454:
3431:
3427:
3418:
3414:
3409:
3406:
3405:
3385:
3382:
3381:
3361:
3357:
3348:
3344:
3336:
3333:
3332:
3312:
3308:
3299:
3295:
3294:
3284:
3280:
3262:
3258:
3251:
3249:
3237:
3233:
3224:
3220:
3212:
3209:
3208:
3188:
3185:
3184:
3180:
3171:
3146:
3143:
3142:
3129:, and the term
3115:
3094:
3091:
3090:
3071:
3068:
3067:
3037:
3034:
3033:
3007:
3003:
2980:
2977:
2976:
2960:
2957:
2956:
2940:
2937:
2936:
2920:
2917:
2916:
2893:
2889:
2885:
2854:
2850:
2846:
2826:
2822:
2798:
2794:
2790:
2786:
2781:
2778:
2777:
2760:
2756:
2747:
2743:
2741:
2738:
2737:
2714:
2710:
2701:
2697:
2695:
2692:
2691:
2663:
2660:
2659:
2655:strictly convex
2637:
2634:
2633:
2613:
2610:
2609:
2591:
2588:
2587:
2561:
2558:
2557:
2538:
2535:
2534:
2509:
2506:
2505:
2474:
2471:
2470:
2448:
2445:
2444:
2423:
2419:
2415:
2384:
2380:
2376:
2368:
2365:
2364:
2331:
2327:
2323:
2318:
2315:
2314:
2284:
2280:
2276:
2271:
2268:
2267:
2251:
2248:
2247:
2231:
2228:
2227:
2211:
2208:
2207:
2188:
2185:
2184:
2168:
2165:
2164:
2130:
2107:
2104:
2103:
2078:
2076:
2073:
2072:
2047:
2043:
2039:
2023:
2019:
2015:
2010:
2007:
2006:
1985:
1981:
1977:
1961:
1957:
1953:
1948:
1945:
1944:
1923:
1919:
1915:
1884:
1880:
1876:
1856:
1852:
1828:
1824:
1820:
1816:
1811:
1808:
1807:
1787:
1783:
1774:
1770:
1768:
1765:
1764:
1745:
1742:
1741:
1719:
1716:
1715:
1694:
1690:
1686:
1670:
1666:
1662:
1657:
1654:
1653:
1632:
1628:
1624:
1608:
1604:
1600:
1595:
1592:
1591:
1572:
1569:
1568:
1552:
1549:
1548:
1522:
1518:
1514:
1502:
1498:
1497:
1493:
1491:
1488:
1487:
1461:
1457:
1453:
1441:
1437:
1436:
1432:
1430:
1427:
1426:
1405:
1401:
1397:
1366:
1362:
1358:
1338:
1334:
1310:
1306:
1302:
1298:
1293:
1290:
1289:
1272:
1268:
1259:
1255:
1253:
1250:
1249:
1226:
1222:
1213:
1209:
1207:
1204:
1203:
1175:
1172:
1171:
1152:
1149:
1148:
1129:
1126:
1125:
1109:
1106:
1105:
1089:
1086:
1085:
1068:
1064:
1062:
1059:
1058:
1041:
1037:
1035:
1032:
1031:
1015:
1012:
1011:
995:
992:
991:
975:
972:
971:
955:
952:
951:
935:
932:
931:
915:
912:
911:
895:
892:
891:
872:
869:
868:
852:
849:
848:
822:
818:
814:
802:
798:
797:
793:
791:
788:
787:
761:
757:
753:
741:
737:
736:
732:
730:
727:
726:
705:
701:
697:
666:
662:
658:
638:
634:
610:
606:
602:
598:
593:
590:
589:
566:
562:
553:
549:
547:
544:
543:
515:
512:
511:
485:
482:
481:
477:be a function.
462:
448:
445:
444:
420:
417:
416:
400:
398:
374:random variable
330:
327:
326:
309:
305:
300:
297:
296:
277:
274:
273:
256:
252:
247:
244:
243:
220:
217:
216:
182:
179:
178:
176:linear function
140:
137:
136:
116:
113:
112:
56:
51:A graph of the
29:
12:
11:
5:
14119:
14109:
14108:
14103:
14098:
14081:
14080:
14078:
14077:
14071:
14069:
14065:
14064:
14062:
14061:
14056:
14054:Strong duality
14051:
14046:
14040:
14038:
14032:
14031:
14029:
14028:
13993:
13991:
13987:
13986:
13984:
13983:
13978:
13969:
13964:
13962:John ellipsoid
13959:
13954:
13949:
13944:
13930:
13924:
13922:
13918:
13917:
13915:
13914:
13909:
13904:
13899:
13894:
13889:
13884:
13879:
13874:
13869:
13864:
13859:
13853:
13851:
13849:results (list)
13844:
13843:
13841:
13840:
13835:
13830:
13825:
13823:Invex function
13820:
13811:
13806:
13801:
13796:
13791:
13785:
13780:
13774:
13772:
13768:
13767:
13765:
13764:
13759:
13754:
13749:
13744:
13739:
13734:
13729:
13724:
13722:Choquet theory
13718:
13716:
13710:
13709:
13707:
13706:
13701:
13696:
13690:
13688:
13687:Basic concepts
13684:
13683:
13672:
13671:
13664:
13657:
13649:
13643:
13642:
13624:
13604:
13603:External links
13601:
13600:
13599:
13585:
13572:
13563:
13551:
13539:
13527:
13518:
13506:
13499:
13490:
13484:
13470:
13467:
13464:
13463:
13450:
13423:
13416:
13387:
13380:
13359:
13341:
13334:
13305:
13298:
13280:
13273:
13247:
13236:xingyuzhou.org
13223:
13214:
13192:
13178:
13158:
13144:
13118:
13108:
13088:
13081:
13061:
13051:
13031:
13025:978-1305266643
13024:
13006:
12988:
12959:
12958:
12956:
12953:
12951:
12950:
12944:
12939:
12934:
12929:
12924:
12914:
12909:
12904:
12902:Invex function
12899:
12894:
12889:
12884:
12879:
12874:
12869:
12864:
12858:
12856:
12853:
12852:
12851:
12840:
12825:
12813:
12810:
12807:
12804:
12799:
12795:
12791:
12788:
12785:
12782:
12779:
12764:
12752:
12749:
12746:
12743:
12740:
12737:
12734:
12731:
12728:
12725:
12722:
12719:
12716:
12713:
12710:
12707:
12687:
12672:
12656:
12653:
12650:
12647:
12644:
12641:
12638:
12627:
12619:
12612:
12611:
12610:
12599:
12596:
12593:
12573:
12570:
12567:
12564:
12561:
12558:
12538:
12535:
12532:
12529:
12526:
12523:
12503:
12500:
12497:
12494:
12491:
12471:
12468:
12465:
12462:
12459:
12456:
12431:
12427:
12423:
12417:
12414:
12411:
12408:
12405:
12394:
12382:
12379:
12376:
12373:
12370:
12367:
12347:
12344:
12341:
12338:
12335:
12315:
12312:
12309:
12306:
12286:
12283:
12280:
12255:
12251:
12247:
12241:
12238:
12235:
12232:
12228:
12225:
12201:
12198:
12192:
12189:
12186:
12183:
12180:
12169:
12157:
12154:
12151:
12148:
12145:
12142:
12139:
12117:
12113:
12109:
12106:
12103:
12100:
12097:
12082:
12070:
12067:
12064:
12061:
12058:
12055:
12052:
12049:
12027:
12022:
12019:
12016:
12013:
12010:
11995:
11984:
11981:
11978:
11954:
11951:
11948:
11928:
11925:
11903:
11899:
11887:
11875:
11872:
11869:
11866:
11863:
11860:
11840:
11837:
11834:
11831:
11828:
11808:
11805:
11802:
11799:
11796:
11793:
11790:
11787:
11784:
11781:
11778:
11775:
11772:
11769:
11766:
11763:
11760:
11757:
11737:
11726:
11714:
11688:
11685:
11682:
11679:
11675:
11671:
11668:
11665:
11662:
11659:
11639:
11636:
11631:
11627:
11623:
11620:
11617:
11614:
11610:
11607:
11584:
11580:
11576:
11573:
11570:
11567:
11564:
11550:
11538:
11535:
11532:
11510:
11505:
11500:
11496:
11492:
11489:
11486:
11483:
11480:
11469:
11457:
11454:
11451:
11426:
11422:
11418:
11414:
11411:
11408:
11405:
11402:
11391:absolute value
11387:
11371:
11368:
11363:
11359:
11355:
11352:
11349:
11346:
11343:
11339:
11336:
11313:
11309:
11305:
11302:
11299:
11296:
11293:
11282:
11266:
11263:
11260:
11257:
11254:
11251:
11248:
11244:
11241:
11218:
11214:
11210:
11207:
11204:
11201:
11198:
11185:
11182:
11180:
11177:
11164:
11139:
11135:
11128:
11125:
11119:
11116:
11113:
11110:
11107:
11087:
11067:
11064:
11061:
11058:
11055:
11052:
11049:
11046:
11043:
11040:
11037:
11034:
11031:
11028:
11025:
11022:
11019:
11016:
11013:
11010:
11007:
11004:
11001:
10998:
10995:
10992:
10989:
10986:
10983:
10980:
10977:
10974:
10971:
10968:
10965:
10962:
10959:
10956:
10953:
10950:
10947:
10944:
10941:
10921:
10918:
10915:
10912:
10909:
10906:
10903:
10900:
10880:
10877:
10874:
10865:that, for all
10854:
10834:
10822:
10819:
10818:
10817:
10810:global minimum
10802:
10758:
10755:
10734:
10731:
10728:
10708:
10705:
10702:
10699:
10696:
10692:
10689:
10668:
10648:
10623:
10619:
10615:
10612:
10607:
10604:
10599:
10596:
10593:
10590:
10587:
10584:
10581:
10557:
10534:
10531:
10528:
10525:
10522:
10517:
10513:
10509:
10506:
10503:
10500:
10497:
10494:
10491:
10488:
10485:
10482:
10479:
10476:
10473:
10470:
10467:
10462:
10458:
10452:
10448:
10444:
10441:
10438:
10435:
10415:
10412:
10409:
10406:
10403:
10400:
10397:
10394:
10391:
10386:
10382:
10376:
10372:
10368:
10365:
10362:
10359:
10354:
10351:
10346:
10343:
10340:
10337:
10334:
10331:
10326:
10322:
10318:
10315:
10312:
10309:
10306:
10303:
10300:
10297:
10294:
10291:
10288:
10285:
10282:
10279:
10259:
10256:
10253:
10250:
10247:
10244:
10241:
10238:
10235:
10232:
10229:
10226:
10223:
10220:
10217:
10214:
10211:
10208:
10205:
10202:
10199:
10175:
10172:
10169:
10166:
10161:
10157:
10133:
10130:
10127:
10124:
10121:
10117:
10114:
10093:
10090:
10087:
10067:
10064:
10061:
10058:
10055:
10051:
10048:
10027:
10024:
10021:
10018:
10014:
10011:
9990:
9987:
9984:
9981:
9976:
9972:
9951:
9948:
9928:
9908:
9905:
9902:
9899:
9894:
9890:
9861:
9858:
9855:
9852:
9849:
9846:
9843:
9838:
9834:
9813:
9802:Hessian matrix
9789:
9784:
9780:
9759:
9739:
9719:
9716:
9713:
9710:
9707:
9704:
9701:
9696:
9692:
9671:
9651:
9628:
9625:
9622:
9602:
9599:
9596:
9576:
9573:
9570:
9567:
9543:
9538:
9534:
9530:
9527:
9524:
9521:
9518:
9515:
9512:
9509:
9506:
9503:
9500:
9495:
9492:
9487:
9484:
9481:
9478:
9475:
9472:
9469:
9466:
9463:
9460:
9457:
9454:
9451:
9448:
9445:
9442:
9439:
9436:
9433:
9430:
9427:
9424:
9421:
9418:
9415:
9412:
9409:
9406:
9403:
9384:
9381:
9378:
9375:
9372:
9369:
9366:
9363:
9343:
9340:
9337:
9317:
9314:
9290:
9285:
9281:
9277:
9274:
9271:
9268:
9263:
9260:
9255:
9252:
9249:
9246:
9243:
9240:
9235:
9231:
9227:
9224:
9221:
9218:
9215:
9212:
9209:
9206:
9203:
9200:
9197:
9194:
9191:
9188:
9157:
9154:
9151:
9127:
9124:
9121:
9118:
9115:
9093:
9089:
9085:
9082:
9079:
9076:
9073:
9070:
9067:
9064:
9061:
9058:
9055:
9052:
9049:
9046:
9043:
9040:
9037:
9034:
9031:
9028:
9025:
9022:
9019:
8997:
8992:
8988:
8984:
8981:
8978:
8975:
8972:
8969:
8966:
8963:
8960:
8957:
8954:
8949:
8945:
8941:
8938:
8935:
8932:
8929:
8926:
8923:
8920:
8917:
8914:
8911:
8908:
8899:in its domain:
8888:
8885:
8882:
8862:
8859:
8856:
8836:
8813:
8810:
8807:
8803:
8800:
8779:
8776:
8773:
8768:
8764:
8760:
8756:
8753:
8743:. Even though
8729:
8726:
8720:
8717:
8712:
8708:
8704:
8700:
8697:
8676:
8671:
8667:
8663:
8643:
8632:
8631:
8620:
8617:
8597:
8594:
8591:
8588:
8585:
8582:
8579:
8575:
8572:
8551:
8541:
8529:
8509:
8506:
8503:
8500:
8497:
8493:
8490:
8469:
8459:
8448:
8445:
8425:
8422:
8419:
8416:
8413:
8409:
8406:
8385:
8362:
8349:
8346:
8345:
8344:
8333:
8330:
8327:
8324:
8321:
8301:
8298:
8295:
8275:
8272:
8269:
8266:
8263:
8243:
8240:
8237:
8234:
8231:
8228:
8225:
8222:
8219:
8216:
8213:
8210:
8207:
8204:
8201:
8198:
8195:
8192:
8189:
8186:
8166:
8163:
8160:
8157:
8154:
8151:
8130:
8126:
8123:
8120:
8117:
8097:
8086:
8073:
8069:
8066:
8063:
8060:
8057:
8054:
8051:
8048:
8045:
8042:
8036:
8033:
8027:
8024:
8021:
8018:
8015:
8012:
8008:
7986:
7980:
7977:
7971:
7967:
7964:
7961:
7958:
7955:
7952:
7949:
7946:
7943:
7923:
7912:
7901:
7898:
7895:
7892:
7889:
7886:
7883:
7880:
7860:
7851:provided that
7840:
7837:
7817:
7814:
7811:
7808:
7805:
7802:
7797:
7794:
7791:
7787:
7783:
7780:
7777:
7774:
7771:
7751:
7748:
7745:
7742:
7739:
7719:
7716:
7713:
7710:
7707:
7704:
7693:
7692:
7691:
7680:
7675:
7670:
7665:
7660:
7656:
7633:
7628:
7623:
7620:
7617:
7612:
7609:
7606:
7601:
7596:
7593:
7573:
7570:
7567:
7564:
7561:
7558:
7555:
7552:
7549:
7546:
7543:
7540:
7518:
7513:
7508:
7503:
7499:
7478:
7467:
7455:
7452:
7449:
7446:
7443:
7440:
7437:
7434:
7431:
7428:
7425:
7422:
7402:
7382:
7371:
7357:
7353:
7330:
7327:
7324:
7321:
7317:
7296:
7276:
7273:
7270:
7267:
7264:
7261:
7258:
7255:
7252:
7249:
7246:
7243:
7223:
7203:
7183:
7169:
7168:
7167:
7155:
7135:
7115:
7112:
7109:
7106:
7103:
7100:
7095:
7092:
7089:
7085:
7081:
7078:
7075:
7072:
7069:
7049:
7029:
7026:
7023:
7020:
7017:
7014:
7000:
6989:
6985:
6981:
6978:
6975:
6970:
6966:
6962:
6959:
6956:
6953:
6950:
6947:
6942:
6938:
6933:
6929:
6926:
6923:
6920:
6917:
6914:
6892:
6888:
6884:
6881:
6878:
6873:
6869:
6845:
6842:
6839:
6836:
6816:
6813:
6810:
6805:
6801:
6795:
6792:
6789:
6785:
6781:
6778:
6775:
6772:
6769:
6747:
6744:
6741:
6737:
6731:
6727:
6723:
6712:
6711:
6710:
6707:
6695:
6690:
6686:
6680:
6676:
6672:
6669:
6666:
6661:
6657:
6651:
6647:
6624:
6620:
6616:
6613:
6610:
6605:
6601:
6580:
6577:
6572:
6568:
6564:
6561:
6558:
6553:
6549:
6534:
6522:
6502:
6499:
6496:
6476:
6465:
6453:
6433:
6430:
6418:
6415:
6414:
6413:
6401:
6398:
6395:
6392:
6389:
6386:
6383:
6363:
6360:
6357:
6354:
6351:
6348:
6345:
6342:
6339:
6336:
6333:
6330:
6327:
6324:
6321:
6318:
6298:
6295:
6275:
6262:A first-order
6260:
6240:
6220:
6217:
6214:
6211:
6208:
6205:
6202:
6199:
6196:
6193:
6190:
6187:
6184:
6181:
6178:
6175:
6172:
6169:
6149:
6146:
6126:
6106:
6092:
6090:
6085:global minimum
6077:
6065:
6060:
6039:
6024:
6021:
6003:
5999:
5996:
5976:
5973:
5970:
5967:
5964:
5961:
5958:
5955:
5952:
5949:
5929:
5926:
5923:
5920:
5917:
5914:
5911:
5908:
5905:
5902:
5879:
5876:
5865:
5854:Hessian matrix
5850:
5849:in the domain.
5838:
5835:
5832:
5823:holds for all
5812:
5809:
5806:
5803:
5800:
5797:
5792:
5788:
5784:
5781:
5778:
5775:
5772:
5769:
5766:
5763:
5760:
5757:
5754:
5751:
5748:
5745:
5725:
5714:
5702:
5699:
5696:
5693:
5690:
5687:
5684:
5678:
5671:
5667:
5664:
5661:
5658:
5655:
5652:
5649:
5646:
5643:
5620:
5617:
5614:
5611:
5608:
5604:
5600:
5597:
5594:
5591:
5588:
5585:
5582:
5570:valued in the
5559:
5556:
5553:
5550:
5547:
5544:
5541:
5538:
5535:
5532:
5521:
5505:
5502:
5499:
5496:
5493:
5490:
5487:
5484:
5481:
5468:
5465:
5464:
5463:
5443:
5438:
5434:
5429:
5425:
5421:
5418:
5415:
5412:
5407:
5403:
5399:
5396:
5390:
5386:
5381:
5375:
5371:
5367:
5362:
5358:
5351:
5346:
5327:
5324:
5319:
5315:
5311:
5306:
5302:
5281:
5261:
5236:
5233:
5230:
5227:
5224:
5221:
5218:
5215:
5212:
5209:
5206:
5203:
5200:
5197:
5194:
5191:
5167:
5164:
5161:
5158:
5155:
5152:
5149:
5146:
5143:
5141:
5139:
5136:
5133:
5130:
5127:
5124:
5121:
5115:
5112:
5109:
5105:
5100:
5097:
5094:
5091:
5088:
5085:
5082:
5076:
5073:
5070:
5066:
5061:
5058:
5056:
5054:
5050:
5043:
5040:
5037:
5033:
5028:
5025:
5022:
5019:
5016:
5012:
5008:
5005:
5001:
4994:
4991:
4988:
4984:
4979:
4976:
4973:
4970:
4967:
4963:
4959:
4956:
4953:
4951:
4949:
4946:
4943:
4940:
4937:
4934:
4931:
4928:
4925:
4922:
4921:
4901:
4896:
4892:
4888:
4885:
4882:
4879:
4876:
4871:
4867:
4863:
4860:
4857:
4833:
4830:
4825:
4821:
4817:
4814:
4811:
4808:
4805:
4803:
4801:
4798:
4795:
4792:
4789:
4786:
4783:
4780:
4777:
4774:
4771:
4768:
4763:
4759:
4755:
4752:
4749:
4746:
4743:
4741:
4739:
4736:
4733:
4730:
4727:
4724:
4721:
4718:
4715:
4712:
4707:
4703:
4699:
4696:
4693:
4690:
4687:
4685:
4683:
4678:
4674:
4670:
4667:
4664:
4661:
4660:
4641:
4638:
4635:
4632:
4629:
4626:
4606:
4603:
4600:
4595:
4591:
4570:
4556:
4555:
4554:
4542:
4522:
4502:
4499:
4496:
4493:
4490:
4487:
4484:
4481:
4478:
4475:
4472:
4469:
4466:
4463:
4460:
4457:
4446:positive reals
4429:
4409:
4406:
4403:
4400:
4397:
4394:
4374:
4363:
4362:
4361:
4349:
4328:
4325:
4304:
4283:
4280:
4259:
4238:
4235:
4214:
4193:
4190:
4164:
4160:
4139:
4136:
4133:
4130:
4108:
4104:
4100:
4097:
4094:
4091:
4088:
4084:
4081:
4058:
4054:
4050:
4047:
4044:
4041:
4038:
4015:
4003:
3983:
3963:
3960:
3957:
3954:
3951:
3948:
3945:
3942:
3938:
3935:
3931:
3928:
3925:
3922:
3919:
3916:
3913:
3910:
3907:
3904:
3889:
3874:differentiable
3870:
3854:
3834:
3814:
3794:
3783:countably many
3779:differentiable
3766:
3738:
3719:
3716:
3692:
3669:
3658:
3644:
3640:
3619:
3614:
3610:
3585:
3580:
3576:
3572:
3567:
3563:
3559:
3556:
3536:
3514:
3510:
3487:
3483:
3462:
3442:
3439:
3434:
3430:
3426:
3421:
3417:
3413:
3389:
3369:
3364:
3360:
3356:
3351:
3347:
3343:
3340:
3315:
3311:
3307:
3302:
3298:
3292:
3287:
3283:
3279:
3276:
3273:
3270:
3265:
3261:
3257:
3254:
3248:
3245:
3240:
3236:
3232:
3227:
3223:
3219:
3216:
3192:
3179:
3176:
3170:
3167:
3150:
3127:concave upward
3114:
3111:
3098:
3078:
3075:
3064:
3058:
3052:is said to be
3041:
3018:
3015:
3010:
3006:
3002:
2999:
2996:
2993:
2990:
2987:
2984:
2964:
2944:
2924:
2901:
2896:
2892:
2888:
2884:
2881:
2878:
2875:
2872:
2869:
2866:
2862:
2857:
2853:
2849:
2845:
2842:
2839:
2835:
2829:
2825:
2821:
2818:
2815:
2812:
2809:
2806:
2801:
2797:
2793:
2789:
2785:
2763:
2759:
2755:
2750:
2746:
2725:
2722:
2717:
2713:
2709:
2704:
2700:
2679:
2676:
2673:
2670:
2667:
2656:
2641:
2621:
2618:
2596:
2585:
2568:
2565:
2545:
2542:
2522:
2519:
2516:
2513:
2493:
2490:
2487:
2484:
2481:
2478:
2458:
2455:
2452:
2431:
2426:
2422:
2418:
2414:
2411:
2408:
2405:
2402:
2399:
2396:
2392:
2387:
2383:
2379:
2375:
2372:
2352:
2349:
2346:
2343:
2339:
2334:
2330:
2326:
2322:
2302:
2299:
2296:
2292:
2287:
2283:
2279:
2275:
2255:
2235:
2215:
2195:
2192:
2172:
2152:
2149:
2146:
2143:
2140:
2137:
2133:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2097:
2081:
2069:
2068:
2055:
2050:
2046:
2042:
2038:
2035:
2031:
2026:
2022:
2018:
2014:
1993:
1988:
1984:
1980:
1976:
1973:
1969:
1964:
1960:
1956:
1952:
1931:
1926:
1922:
1918:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1892:
1887:
1883:
1879:
1875:
1872:
1869:
1865:
1859:
1855:
1851:
1848:
1845:
1842:
1839:
1836:
1831:
1827:
1823:
1819:
1815:
1795:
1790:
1786:
1782:
1777:
1773:
1752:
1749:
1729:
1726:
1723:
1702:
1697:
1693:
1689:
1685:
1682:
1678:
1673:
1669:
1665:
1661:
1640:
1635:
1631:
1627:
1623:
1620:
1616:
1611:
1607:
1603:
1599:
1579:
1576:
1556:
1535:
1530:
1525:
1521:
1517:
1513:
1510:
1505:
1501:
1496:
1474:
1469:
1464:
1460:
1456:
1452:
1449:
1444:
1440:
1435:
1413:
1408:
1404:
1400:
1396:
1393:
1390:
1387:
1384:
1381:
1378:
1374:
1369:
1365:
1361:
1357:
1354:
1351:
1347:
1341:
1337:
1333:
1330:
1327:
1324:
1321:
1318:
1313:
1309:
1305:
1301:
1297:
1275:
1271:
1267:
1262:
1258:
1237:
1234:
1229:
1225:
1221:
1216:
1212:
1191:
1188:
1185:
1182:
1179:
1168:
1156:
1136:
1133:
1113:
1093:
1071:
1067:
1044:
1040:
1019:
999:
979:
959:
950:or decreasing
939:
919:
899:
879:
876:
856:
835:
830:
825:
821:
817:
813:
810:
805:
801:
796:
774:
769:
764:
760:
756:
752:
749:
744:
740:
735:
713:
708:
704:
700:
696:
693:
690:
687:
684:
681:
678:
674:
669:
665:
661:
657:
654:
651:
647:
641:
637:
633:
630:
627:
624:
621:
618:
613:
609:
605:
601:
597:
577:
574:
569:
565:
561:
556:
552:
531:
528:
525:
522:
519:
504:
489:
465:
461:
458:
455:
452:
424:
397:
394:
370:expected value
334:
312:
308:
304:
281:
259:
255:
251:
224:
204:
201:
198:
195:
192:
189:
186:
164:if and only if
160:differentiable
144:
120:
27:
9:
6:
4:
3:
2:
14118:
14107:
14104:
14102:
14099:
14097:
14094:
14093:
14091:
14076:
14073:
14072:
14070:
14066:
14060:
14057:
14055:
14052:
14050:
14047:
14045:
14042:
14041:
14039:
14037:
14033:
14026:
14024:
14018:
14016:
14010:
14006:
14002:
13998:
13995:
13994:
13992:
13988:
13982:
13979:
13977:
13973:
13970:
13968:
13965:
13963:
13960:
13958:
13955:
13953:
13950:
13948:
13945:
13943:
13939:
13935:
13931:
13929:
13926:
13925:
13923:
13919:
13913:
13910:
13908:
13905:
13903:
13900:
13898:
13895:
13893:
13892:Mazur's lemma
13890:
13888:
13885:
13883:
13880:
13878:
13875:
13873:
13870:
13868:
13865:
13863:
13860:
13858:
13855:
13854:
13852:
13850:
13845:
13839:
13838:Subderivative
13836:
13834:
13831:
13829:
13826:
13824:
13821:
13819:
13815:
13812:
13810:
13807:
13805:
13802:
13800:
13797:
13795:
13792:
13790:
13786:
13784:
13781:
13779:
13776:
13775:
13773:
13769:
13763:
13760:
13758:
13755:
13753:
13750:
13748:
13745:
13743:
13740:
13738:
13735:
13733:
13730:
13728:
13725:
13723:
13720:
13719:
13717:
13715:
13714:Topics (list)
13711:
13705:
13702:
13700:
13697:
13695:
13692:
13691:
13689:
13685:
13681:
13677:
13670:
13665:
13663:
13658:
13656:
13651:
13650:
13647:
13639:
13635:
13634:
13629:
13625:
13621:
13617:
13616:
13611:
13607:
13606:
13596:
13592:
13588:
13586:981-238-067-1
13582:
13578:
13573:
13569:
13564:
13560:
13556:
13552:
13548:
13544:
13540:
13536:
13532:
13528:
13524:
13519:
13515:
13511:
13507:
13504:
13500:
13496:
13491:
13488:
13485:
13481:
13477:
13473:
13472:
13460:
13454:
13446:
13442:
13438:
13434:
13427:
13419:
13413:
13409:
13404:
13403:
13394:
13392:
13383:
13377:
13373:
13366:
13364:
13352:
13345:
13337:
13335:9781402075537
13331:
13327:
13322:
13321:
13312:
13310:
13301:
13299:9780521339841
13295:
13291:
13284:
13276:
13274:9781886529458
13270:
13266:
13261:
13260:
13251:
13237:
13233:
13227:
13218:
13202:
13196:
13181:
13179:9780122206504
13175:
13171:
13170:
13162:
13147:
13141:
13134:
13133:
13125:
13123:
13116:
13111:
13105:
13101:
13100:
13092:
13084:
13078:
13074:
13073:
13065:
13059:
13054:
13048:
13044:
13043:
13035:
13027:
13021:
13017:
13010:
13002:
12998:
12992:
12977:
12970:
12964:
12960:
12948:
12947:Subderivative
12945:
12943:
12940:
12938:
12935:
12933:
12930:
12928:
12925:
12922:
12918:
12915:
12913:
12910:
12908:
12905:
12903:
12900:
12898:
12895:
12893:
12890:
12888:
12885:
12883:
12880:
12878:
12875:
12873:
12870:
12868:
12865:
12863:
12860:
12859:
12849:
12845:
12841:
12838:
12834:
12830:
12826:
12811:
12808:
12805:
12802:
12797:
12793:
12789:
12783:
12777:
12769:
12765:
12747:
12741:
12738:
12732:
12726:
12723:
12717:
12714:
12711:
12705:
12685:
12677:
12673:
12670:
12651:
12642:
12639:
12636:
12629:The function
12628:
12625:
12622:
12621:
12617:
12614:Functions of
12597:
12594:
12591:
12565:
12559:
12533:
12530:
12524:
12495:
12492:
12466:
12460:
12454:
12429:
12425:
12421:
12415:
12409:
12403:
12396:The function
12395:
12377:
12374:
12368:
12339:
12336:
12310:
12304:
12284:
12281:
12278:
12253:
12249:
12245:
12239:
12233:
12226:
12223:
12199:
12196:
12190:
12184:
12178:
12171:The function
12170:
12155:
12152:
12149:
12143:
12137:
12115:
12111:
12107:
12101:
12095:
12087:
12083:
12068:
12065:
12062:
12059:
12053:
12047:
12025:
12020:
12014:
12008:
12000:
11996:
11982:
11979:
11976:
11968:
11952:
11949:
11946:
11926:
11923:
11901:
11897:
11889:The function
11888:
11873:
11867:
11864:
11861:
11838:
11835:
11832:
11829:
11826:
11806:
11803:
11797:
11791:
11788:
11785:
11782:
11776:
11770:
11767:
11761:
11755:
11735:
11728:The function
11727:
11712:
11704:
11683:
11677:
11673:
11669:
11663:
11657:
11637:
11634:
11629:
11625:
11621:
11615:
11608:
11605:
11582:
11578:
11574:
11568:
11562:
11555:
11551:
11536:
11533:
11530:
11508:
11498:
11490:
11484:
11478:
11471:The function
11470:
11455:
11452:
11449:
11441:
11420:
11412:
11406:
11400:
11392:
11388:
11369:
11366:
11361:
11357:
11353:
11350:
11344:
11337:
11334:
11311:
11307:
11303:
11297:
11291:
11284:The function
11283:
11264:
11261:
11258:
11255:
11249:
11242:
11239:
11216:
11212:
11208:
11202:
11196:
11189:The function
11188:
11187:
11176:
11162:
11153:
11137:
11133:
11126:
11123:
11117:
11111:
11105:
11085:
11059:
11056:
11053:
11044:
11038:
11035:
11032:
11026:
11023:
11017:
11011:
11005:
11002:
10999:
10993:
10987:
10981:
10978:
10975:
10969:
10963:
10960:
10957:
10951:
10948:
10945:
10939:
10919:
10913:
10910:
10907:
10901:
10898:
10878:
10875:
10872:
10852:
10832:
10815:
10811:
10808:has a unique
10807:
10804:The function
10803:
10800:
10796:
10792:
10788:
10784:
10780:
10776:
10772:
10771:
10770:
10768:
10764:
10754:
10750:
10748:
10732:
10729:
10726:
10706:
10703:
10697:
10690:
10687:
10666:
10646:
10637:
10621:
10613:
10605:
10602:
10597:
10591:
10585:
10579:
10571:
10555:
10546:
10529:
10526:
10523:
10515:
10507:
10504:
10501:
10495:
10492:
10486:
10483:
10480:
10471:
10465:
10460:
10450:
10442:
10439:
10436:
10410:
10407:
10404:
10395:
10389:
10384:
10374:
10366:
10363:
10360:
10352:
10349:
10344:
10338:
10335:
10332:
10324:
10316:
10310:
10304:
10298:
10292:
10289:
10283:
10277:
10251:
10248:
10245:
10239:
10236:
10233:
10230:
10224:
10221:
10218:
10212:
10209:
10206:
10200:
10197:
10190:there exists
10189:
10170:
10164:
10159:
10145:
10131:
10128:
10122:
10115:
10112:
10091:
10088:
10085:
10065:
10062:
10056:
10049:
10046:
10025:
10019:
10012:
10009:
9985:
9979:
9974:
9949:
9946:
9926:
9903:
9897:
9892:
9879:
9875:
9859:
9856:
9853:
9847:
9841:
9836:
9811:
9803:
9787:
9782:
9757:
9737:
9717:
9714:
9711:
9705:
9699:
9694:
9669:
9649:
9640:
9626:
9623:
9620:
9600:
9597:
9594:
9574:
9571:
9565:
9556:
9541:
9536:
9528:
9525:
9522:
9513:
9510:
9507:
9501:
9498:
9493:
9490:
9485:
9479:
9473:
9467:
9464:
9461:
9455:
9449:
9443:
9440:
9437:
9431:
9425:
9422:
9419:
9413:
9410:
9407:
9401:
9382:
9376:
9373:
9370:
9364:
9361:
9341:
9338:
9335:
9315:
9312:
9303:
9288:
9283:
9275:
9272:
9269:
9261:
9258:
9253:
9247:
9244:
9241:
9233:
9225:
9219:
9213:
9207:
9201:
9198:
9192:
9186:
9177:
9175:
9171:
9152:
9141:
9140:inner product
9122:
9119:
9116:
9091:
9083:
9080:
9077:
9071:
9068:
9062:
9059:
9056:
9053:
9047:
9041:
9035:
9029:
9023:
8995:
8990:
8982:
8979:
8976:
8970:
8967:
8961:
8958:
8955:
8947:
8936:
8930:
8924:
8918:
8912:
8886:
8883:
8880:
8860:
8857:
8854:
8834:
8825:
8808:
8801:
8798:
8777:
8774:
8766:
8762:
8754:
8751:
8727:
8724:
8718:
8710:
8706:
8698:
8695:
8669:
8665:
8641:
8618:
8615:
8595:
8592:
8589:
8586:
8580:
8573:
8570:
8549:
8542:
8527:
8507:
8504:
8498:
8491:
8488:
8467:
8460:
8446:
8443:
8423:
8420:
8414:
8407:
8404:
8383:
8376:
8375:
8374:
8360:
8331:
8328:
8325:
8322:
8319:
8312:that satisfy
8299:
8296:
8293:
8273:
8270:
8267:
8264:
8261:
8238:
8232:
8229:
8226:
8220:
8214:
8211:
8208:
8202:
8199:
8196:
8193:
8190:
8184:
8164:
8161:
8155:
8149:
8121:
8118:
8115:
8095:
8087:
8071:
8067:
8064:
8061:
8058:
8052:
8046:
8043:
8040:
8034:
8031:
8025:
8019:
8016:
8013:
8006:
7984:
7978:
7975:
7969:
7965:
7962:
7959:
7953:
7950:
7947:
7941:
7921:
7913:
7899:
7893:
7890:
7884:
7878:
7858:
7838:
7835:
7828:is convex in
7812:
7809:
7806:
7800:
7795:
7792:
7789:
7781:
7775:
7769:
7746:
7743:
7740:
7730:is convex in
7714:
7711:
7708:
7702:
7694:
7678:
7673:
7663:
7658:
7654:
7631:
7621:
7618:
7615:
7610:
7607:
7604:
7594:
7591:
7568:
7565:
7562:
7559:
7553:
7550:
7544:
7538:
7531:, then so is
7516:
7506:
7501:
7497:
7476:
7468:
7447:
7441:
7435:
7432:
7426:
7420:
7400:
7380:
7372:
7355:
7351:
7325:
7319:
7315:
7294:
7268:
7262:
7256:
7253:
7247:
7241:
7221:
7201:
7181:
7173:
7172:
7171:Composition:
7170:
7153:
7133:
7126:is convex in
7110:
7107:
7104:
7098:
7093:
7090:
7087:
7079:
7073:
7067:
7047:
7040:is convex in
7024:
7021:
7018:
7012:
7004:
7001:
6987:
6983:
6976:
6968:
6964:
6960:
6957:
6954:
6948:
6940:
6936:
6931:
6924:
6918:
6912:
6890:
6886:
6882:
6879:
6876:
6871:
6867:
6858:
6857:
6840:
6834:
6811:
6803:
6799:
6793:
6790:
6787:
6779:
6773:
6767:
6745:
6742:
6739:
6729:
6725:
6713:
6708:
6693:
6688:
6684:
6678:
6674:
6670:
6667:
6664:
6659:
6655:
6649:
6645:
6622:
6618:
6614:
6611:
6608:
6603:
6599:
6578:
6575:
6570:
6566:
6562:
6559:
6556:
6551:
6547:
6538:
6537:
6535:
6520:
6500:
6497:
6494:
6474:
6466:
6451:
6431:
6428:
6421:
6420:
6399:
6396:
6393:
6390:
6387:
6384:
6381:
6358:
6355:
6352:
6346:
6343:
6340:
6334:
6331:
6328:
6325:
6322:
6316:
6296:
6293:
6273:
6265:
6261:
6258:
6254:
6218:
6209:
6203:
6194:
6191:
6182:
6176:
6170:
6147:
6144:
6124:
6104:
6096:
6093:
6088:
6086:
6082:
6081:local minimum
6078:
6063:
6058:
6037:
6029:
6025:
6022:
6020:
6017:
5997:
5994:
5971:
5968:
5962:
5956:
5953:
5950:
5924:
5921:
5915:
5909:
5906:
5903:
5893:
5892:sublevel sets
5877:
5874:
5866:
5863:
5859:
5855:
5851:
5836:
5833:
5830:
5807:
5804:
5801:
5795:
5790:
5782:
5776:
5770:
5764:
5758:
5755:
5749:
5743:
5723:
5715:
5694:
5688:
5685:
5682:
5676:
5665:
5662:
5659:
5653:
5650:
5647:
5634:
5612:
5606:
5598:
5589:
5583:
5573:
5551:
5545:
5536:
5533:
5530:
5522:
5519:
5503:
5500:
5497:
5491:
5488:
5485:
5479:
5471:
5470:
5461:
5457:
5441:
5436:
5427:
5423:
5416:
5413:
5405:
5401:
5394:
5388:
5384:
5379:
5373:
5369:
5365:
5360:
5356:
5349:
5344:
5325:
5322:
5317:
5313:
5309:
5304:
5300:
5279:
5259:
5251:
5250:
5248:
5231:
5228:
5225:
5219:
5216:
5210:
5204:
5201:
5195:
5189:
5165:
5159:
5156:
5153:
5147:
5144:
5142:
5131:
5128:
5125:
5119:
5113:
5110:
5107:
5103:
5098:
5092:
5089:
5086:
5080:
5074:
5071:
5068:
5064:
5059:
5057:
5048:
5041:
5038:
5035:
5031:
5023:
5020:
5017:
5010:
5006:
5003:
4999:
4992:
4989:
4986:
4982:
4974:
4971:
4968:
4961:
4957:
4954:
4952:
4944:
4938:
4935:
4929:
4923:
4894:
4890:
4883:
4880:
4877:
4869:
4865:
4861:
4855:
4831:
4823:
4819:
4812:
4809:
4806:
4804:
4793:
4787:
4781:
4778:
4775:
4769:
4761:
4757:
4750:
4747:
4744:
4742:
4731:
4728:
4722:
4719:
4716:
4710:
4705:
4701:
4697:
4691:
4688:
4686:
4676:
4672:
4668:
4662:
4639:
4636:
4633:
4630:
4627:
4624:
4604:
4601:
4598:
4593:
4589:
4568:
4540:
4520:
4497:
4491:
4488:
4482:
4476:
4473:
4467:
4464:
4461:
4455:
4447:
4443:
4442:superadditive
4427:
4407:
4404:
4398:
4392:
4372:
4364:
4347:
4326:
4323:
4302:
4281:
4278:
4257:
4236:
4233:
4212:
4191:
4188:
4179:
4178:
4162:
4158:
4137:
4134:
4131:
4128:
4106:
4102:
4098:
4095:
4089:
4082:
4079:
4056:
4052:
4048:
4042:
4036:
4028:
4024:
4020:
4016:
4001:
3981:
3958:
3955:
3952:
3943:
3936:
3933:
3929:
3923:
3917:
3914:
3908:
3902:
3894:
3890:
3887:
3883:
3879:
3875:
3871:
3868:
3852:
3832:
3812:
3792:
3784:
3780:
3764:
3756:
3752:
3736:
3717:
3714:
3706:
3690:
3683:
3682:open interval
3667:
3659:
3642:
3638:
3617:
3612:
3608:
3599:
3578:
3574:
3570:
3565:
3561:
3554:
3534:
3512:
3508:
3485:
3481:
3460:
3440:
3432:
3428:
3424:
3419:
3415:
3403:
3387:
3362:
3358:
3354:
3349:
3345:
3338:
3313:
3309:
3305:
3300:
3296:
3285:
3281:
3274:
3271:
3263:
3259:
3252:
3246:
3238:
3234:
3230:
3225:
3221:
3214:
3206:
3190:
3182:
3181:
3175:
3166:
3164:
3148:
3140:
3139:convex upward
3136:
3132:
3128:
3124:
3120:
3110:
3096:
3076:
3073:
3065:
3062:
3059:
3057:
3054:
3039:
3032:The function
3030:
3016:
3013:
3008:
3004:
3000:
2994:
2991:
2988:
2982:
2962:
2942:
2922:
2913:
2899:
2894:
2890:
2886:
2882:
2876:
2873:
2870:
2864:
2860:
2855:
2851:
2847:
2843:
2840:
2837:
2833:
2827:
2823:
2816:
2813:
2810:
2804:
2799:
2795:
2791:
2787:
2783:
2761:
2757:
2753:
2748:
2744:
2723:
2720:
2715:
2711:
2707:
2702:
2698:
2677:
2674:
2671:
2668:
2665:
2657:
2654:
2639:
2619:
2616:
2594:
2583:
2580:
2579:as a value.
2563:
2540:
2517:
2511:
2485:
2479:
2476:
2453:
2450:
2429:
2424:
2420:
2416:
2412:
2406:
2403:
2400:
2394:
2390:
2385:
2381:
2377:
2373:
2370:
2350:
2344:
2341:
2337:
2332:
2328:
2324:
2320:
2297:
2294:
2290:
2285:
2281:
2277:
2273:
2253:
2233:
2213:
2190:
2170:
2150:
2141:
2135:
2127:
2118:
2112:
2102:
2098:
2095:
2053:
2048:
2044:
2040:
2036:
2033:
2029:
2024:
2020:
2016:
2012:
1991:
1986:
1982:
1978:
1974:
1971:
1967:
1962:
1958:
1954:
1950:
1929:
1924:
1920:
1916:
1912:
1906:
1903:
1900:
1894:
1890:
1885:
1881:
1877:
1873:
1870:
1867:
1863:
1857:
1853:
1846:
1843:
1840:
1834:
1829:
1825:
1821:
1817:
1813:
1793:
1788:
1784:
1780:
1775:
1771:
1750:
1747:
1727:
1724:
1721:
1700:
1695:
1691:
1687:
1683:
1680:
1676:
1671:
1667:
1663:
1659:
1638:
1633:
1629:
1625:
1621:
1618:
1614:
1609:
1605:
1601:
1597:
1577:
1574:
1554:
1533:
1528:
1523:
1519:
1515:
1511:
1508:
1503:
1499:
1494:
1472:
1467:
1462:
1458:
1454:
1450:
1447:
1442:
1438:
1433:
1411:
1406:
1402:
1398:
1394:
1388:
1385:
1382:
1376:
1372:
1367:
1363:
1359:
1355:
1352:
1349:
1345:
1339:
1335:
1328:
1325:
1322:
1316:
1311:
1307:
1303:
1299:
1295:
1273:
1269:
1265:
1260:
1256:
1235:
1232:
1227:
1223:
1219:
1214:
1210:
1189:
1186:
1183:
1180:
1177:
1169:
1154:
1134:
1131:
1111:
1091:
1069:
1065:
1042:
1038:
1017:
997:
977:
957:
937:
917:
897:
877:
874:
854:
833:
828:
823:
819:
815:
811:
808:
803:
799:
794:
772:
767:
762:
758:
754:
750:
747:
742:
738:
733:
711:
706:
702:
698:
694:
688:
685:
682:
676:
672:
667:
663:
659:
655:
652:
649:
645:
639:
635:
628:
625:
622:
616:
611:
607:
603:
599:
595:
575:
572:
567:
563:
559:
554:
550:
529:
526:
523:
520:
517:
509:
507:
505:
502:
487:
478:
456:
453:
450:
442:
438:
437:convex subset
422:
393:
391:
387:
383:
379:
375:
371:
367:
363:
359:
355:
351:
346:
332:
310:
306:
302:
295:
279:
257:
253:
249:
242:
238:
222:
202:
199:
196:
190:
184:
177:
173:
169:
165:
161:
156:
142:
134:
118:
110:
106:
105:
100:
96:
92:
88:
84:
75:
67:
63:
59:
54:
49:
42:
38:
33:
23:
18:
14059:Weak duality
14022:
14014:
13934:Orthogonally
13817:
13698:
13631:
13613:
13576:
13570:. CRC Press.
13567:
13558:
13546:
13534:
13522:
13513:
13494:
13479:
13453:
13436:
13432:
13426:
13401:
13371:
13344:
13319:
13289:
13283:
13258:
13250:
13239:. Retrieved
13235:
13226:
13217:
13205:. Retrieved
13195:
13183:. Retrieved
13168:
13161:
13149:. Retrieved
13131:
13098:
13091:
13071:
13064:
13041:
13034:
13015:
13009:
12991:
12979:. Retrieved
12975:
12963:
12921:monotonicity
12877:Convex curve
12615:
11154:
10824:
10813:
10805:
10794:
10790:
10786:
10782:
10774:
10766:
10762:
10760:
10751:
10638:
10569:
10547:
10146:
9919:be at least
9641:
9557:
9304:
9178:
8826:
8633:
8351:
7998:with domain
7646:with domain
6251:denotes the
6016:
4560:
3172:
3138:
3135:concave down
3134:
3126:
3122:
3118:
3116:
3061:
3053:
3031:
2914:
2653:
2581:
2094:
2070:
501:
479:
441:vector space
414:
384:such as the
382:inequalities
350:optimization
347:
157:
103:
95:line segment
90:
80:
65:
61:
57:
14049:Duality gap
14044:Dual system
13928:Convex hull
13439:: 283–284.
13151:October 15,
10636:is convex.
10548:A function
9824:means that
9176:functions.
5523:A function
5292:if for all
5252:A function
3453:means that
3331:(note that
3123:convex down
890:increasing
237:real number
83:mathematics
14090:Categories
13972:Radial set
13942:Convex set
13704:Convex set
13469:References
13381:9812380671
13241:2023-09-27
13185:August 29,
12671:is convex.
11549:is convex.
10932:satisfies
10270:such that
9878:eigenvalue
8687:such that
8085:is convex.
7466:is convex.
6533:is convex.
6464:is convex.
5856:of second
5460:Sierpiński
4448:, that is
3878:derivative
3705:continuous
3169:Properties
2736:such that
2652:is called
1248:such that
500:is called
439:of a real
396:Definition
109:convex set
89:is called
41:convex set
13957:Hypograph
13638:EMS Press
13620:EMS Press
12643:
12637:−
12624:LogSumExp
12618:variables
12569:∞
12563:∞
12560:−
12528:∞
12525:−
12499:∞
12470:∞
12372:∞
12369:−
12343:∞
12153:−
12066:
11980:≤
11950:≥
11534:≥
11393:function
11367:≥
11163:ϕ
11134:α
11112:α
11106:ϕ
11086:ϕ
11063:‖
11057:−
11051:‖
11045:ϕ
11036:−
11024:−
11003:−
10976:≤
10961:−
10902:∈
10833:ϕ
10779:level set
10730:∈
10618:‖
10611:‖
10598:−
10583:↦
10527:−
10505:−
10493:≥
10484:−
10457:∇
10440:−
10408:−
10381:∇
10364:−
10336:−
10308:∇
10240:∈
10222:−
10201:∈
10156:∇
10129:≥
10063:≥
9971:∇
9889:∇
9854:−
9833:∇
9812:⪰
9779:∇
9712:⪰
9691:∇
9569:→
9533:‖
9526:−
9520:‖
9511:−
9486:−
9465:−
9438:≤
9423:−
9365:∈
9280:‖
9273:−
9267:‖
9245:−
9217:∇
9199:≥
9156:‖
9153:⋅
9150:‖
9126:⟩
9123:⋅
9117:⋅
9114:⟨
9088:‖
9081:−
9075:‖
9069:≥
9066:⟩
9060:−
9039:∇
9036:−
9021:∇
9018:⟨
8987:‖
8980:−
8974:‖
8968:≥
8959:−
8928:∇
8925:−
8910:∇
8587:≥
8421:≥
8329:≤
8271:∈
8209:≤
8162:≤
8125:→
8047:
8041:∈
7897:∞
7894:−
7891:≠
7793:∈
7664:⊆
7622:∈
7608:×
7595:∈
7507:⊆
7091:∈
6958:…
6880:…
6791:∈
6743:∈
6668:⋯
6612:…
6576:≥
6560:…
6429:−
6204:
6192:≥
6171:
6076:- convex.
5998:∈
5969:≤
5805:−
5796:⋅
5774:∇
5756:≥
5686:≥
5666:×
5660:∈
5616:∞
5613:±
5607:∪
5593:∞
5587:∞
5584:−
5555:∞
5549:∞
5546:−
5540:→
5389:≤
5323:∈
5217:≤
5060:≤
4878:≤
4807:≤
4779:−
4745:≤
4729:⋅
4720:−
4634:≤
4628:≤
4474:≥
4405:≤
3956:−
3915:≥
3402:symmetric
3306:−
3272:−
3149:∪
3117:The term
3074:−
2874:−
2814:−
2754:≠
2721:∈
2595:≤
2567:∞
2544:∞
2541:−
2521:∞
2515:∞
2512:−
2489:∞
2486:−
2480:⋅
2457:∞
2454:⋅
2404:−
2348:∞
2345:±
2301:∞
2298:±
2194:∞
2191:±
2145:∞
2142:±
2136:∪
2122:∞
2116:∞
2113:−
2034:≤
1972:≤
1904:−
1868:≤
1844:−
1681:≤
1619:≤
1386:−
1350:≤
1326:−
1266:≠
1233:∈
686:−
650:≤
626:−
573:∈
527:≤
521:≤
460:→
143:∩
119:∪
53:bivariate
13981:Zonotope
13952:Epigraph
13557:(1970).
13545:(1969).
13533:(1984).
13478:(2003).
13016:Calculus
13001:Archived
12855:See also
12227:″
12088:include
11609:″
11338:″
11243:″
11179:Examples
10769:, then:
10719:for all
10691:″
10116:″
10050:″
10013:″
9939:for all
9730:for all
9174:elliptic
8802:″
8755:″
8699:″
8608:for all
8574:″
8520:for all
8492:″
8436:for all
8408:″
8254:for any
7584:, where
7343:because
7146:even if
6089:strictly
5633:epigraph
5182:Namely,
4327:″
4282:′
4237:′
4192:″
4083:″
4027:converse
3974:for all
3937:′
3893:tangents
3183:Suppose
2690:and all
2226:to take
1943:because
1202:and all
1170:For all
542:and all
510:For all
443:and let
354:open set
158:A twice-
104:epigraph
22:interval
14036:Duality
13938:Pseudo-
13912:Ursescu
13809:Pseudo-
13783:Concave
13762:Simplex
13742:Duality
13640:, 2001
13622:, 2001
13595:1921556
12981:3 March
11967:concave
10799:compact
9800:is the
9138:is any
4444:on the
4420:, then
3749:admits
3131:concave
3060:(resp.
3056:concave
1104:or the
358:minimum
215:(where
93:if the
14019:, and
13990:Series
13907:Simons
13814:Quasi-
13804:Proper
13789:Closed
13593:
13583:
13414:
13378:
13332:
13296:
13271:
13207:14 May
13176:
13142:
13106:
13079:
13049:
13022:
12827:Every
11078:where
10777:, the
9142:, and
9106:where
6231:where
6059:argmin
5680:
5674:
4561:Since
3119:convex
503:convex
172:domain
91:convex
13847:Main
13354:(PDF)
13328:–64.
13136:(pdf)
12972:(PDF)
12955:Notes
12846:of a
12447:with
11382:, so
11277:, so
10797:} is
10426:Then
10078:. If
7762:then
7060:then
7005:: If
6160:then
6117:. If
6087:. A
5987:with
4848:From
4558:Proof
4225:then
3066:) if
970:from
910:from
480:Then
435:be a
372:of a
364:. In
239:), a
235:is a
37:graph
13967:Lens
13921:Sets
13771:Maps
13678:and
13581:ISBN
13412:ISBN
13376:ISBN
13330:ISBN
13294:ISBN
13269:ISBN
13209:2016
13187:2012
13174:ISBN
13153:2011
13140:ISBN
13104:ISBN
13077:ISBN
13047:ISBN
13020:ISBN
12983:2017
12842:The
12835:and
12829:norm
12282:>
12215:has
12130:and
12040:and
11965:and
11836:<
11830:<
11819:for
11635:>
11552:The
11523:for
11389:The
11326:has
11262:>
11231:has
10793:) ≤
10704:>
9624:>
9170:norm
8858:>
8775:>
8593:>
8505:>
8088:Let
8065:>
7194:and
6591:and
6397:>
6286:and
6079:Any
5940:and
5922:<
5890:the
4533:and
4150:but
3994:and
3500:and
3205:real
2838:<
2675:<
2669:<
2617:<
2556:and
2469:and
2005:and
1486:and
1187:<
1181:<
1057:and
786:and
415:Let
388:and
166:its
85:, a
14021:(Hw
13441:doi
13408:144
12646:det
12640:log
12297:so
12063:log
11705:if
11701:is
10812:on
10761:If
9880:of
9872:is
8044:Dom
7914:If
7786:inf
7373:If
7174:If
7084:sup
6928:max
6859:If
6784:sup
6539:if
6467:If
5860:is
5516:is
4440:is
4365:If
4071:is
3880:is
3777:is
3707:on
3703:is
3600:in
3596:is
3527:).
3404:in
3400:is
3137:or
3125:or
2313:or
2246:or
1763:or
1740:or
1714:at
1652:or
1084:in
990:to
930:to
81:In
14092::
14013:(H
14011:,
14007:,
14003:,
13940:)
13936:,
13816:)
13794:K-
13636:,
13630:,
13618:,
13612:,
13591:MR
13589:.
13437:12
13435:.
13410:.
13390:^
13362:^
13326:63
13308:^
13267:.
13265:72
13234:.
13121:^
12999:.
12974:.
12598:0.
11983:0.
11456:0.
11354:12
10785:|
9601:0.
8332:1.
5247:.
4099:12
3895::
3872:A
3869:).
2776::
1288::
588::
392:.
155:.
64:+
62:xy
60:+
14027:)
14025:)
14023:x
14017:)
14015:x
13999:(
13974:/
13932:(
13787:(
13668:e
13661:t
13654:v
13597:.
13447:.
13443::
13420:.
13384:.
13356:.
13338:.
13302:.
13277:.
13244:.
13211:.
13189:.
13155:.
13112:.
13085:.
13055:.
13028:.
12985:.
12839:.
12812:,
12809:b
12806:+
12803:x
12798:T
12794:a
12790:=
12787:)
12784:x
12781:(
12778:f
12751:)
12748:b
12745:(
12742:f
12739:+
12736:)
12733:a
12730:(
12727:f
12724:=
12721:)
12718:b
12715:+
12712:a
12709:(
12706:f
12686:f
12655:)
12652:X
12649:(
12616:n
12595:=
12592:x
12572:)
12566:,
12557:(
12537:)
12534:0
12531:,
12522:(
12502:)
12496:,
12493:0
12490:(
12467:=
12464:)
12461:0
12458:(
12455:f
12430:2
12426:x
12422:1
12416:=
12413:)
12410:x
12407:(
12404:f
12393:.
12381:)
12378:0
12375:,
12366:(
12346:)
12340:,
12337:0
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12314:)
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12279:x
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12250:x
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12240:=
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12234:x
12231:(
12224:f
12200:x
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12191:=
12188:)
12185:x
12182:(
12179:f
12168:.
12156:x
12150:=
12147:)
12144:x
12141:(
12138:k
12116:2
12112:x
12108:=
12105:)
12102:x
12099:(
12096:h
12081:.
12069:x
12060:=
12057:)
12054:x
12051:(
12048:g
12026:x
12021:=
12018:)
12015:x
12012:(
12009:f
11977:x
11953:0
11947:x
11927:x
11924:6
11902:3
11898:x
11874:,
11871:)
11868:1
11865:,
11862:0
11859:(
11839:1
11833:x
11827:0
11807:0
11804:=
11801:)
11798:x
11795:(
11792:f
11789:,
11786:1
11783:=
11780:)
11777:1
11774:(
11771:f
11768:=
11765:)
11762:0
11759:(
11756:f
11736:f
11713:f
11687:)
11684:x
11681:(
11678:f
11674:e
11670:=
11667:)
11664:x
11661:(
11658:g
11638:0
11630:x
11626:e
11622:=
11619:)
11616:x
11613:(
11606:f
11583:x
11579:e
11575:=
11572:)
11569:x
11566:(
11563:f
11537:1
11531:p
11509:p
11504:|
11499:x
11495:|
11491:=
11488:)
11485:x
11482:(
11479:f
11453:=
11450:x
11425:|
11421:x
11417:|
11413:=
11410:)
11407:x
11404:(
11401:f
11384:f
11370:0
11362:2
11358:x
11351:=
11348:)
11345:x
11342:(
11335:f
11312:4
11308:x
11304:=
11301:)
11298:x
11295:(
11292:f
11279:f
11265:0
11259:2
11256:=
11253:)
11250:x
11247:(
11240:f
11217:2
11213:x
11209:=
11206:)
11203:x
11200:(
11197:f
11138:2
11127:2
11124:m
11118:=
11115:)
11109:(
11066:)
11060:y
11054:x
11048:(
11042:)
11039:t
11033:1
11030:(
11027:t
11021:)
11018:y
11015:(
11012:f
11009:)
11006:t
11000:1
10997:(
10994:+
10991:)
10988:x
10985:(
10982:f
10979:t
10973:)
10970:y
10967:)
10964:t
10958:1
10955:(
10952:+
10949:x
10946:t
10943:(
10940:f
10920:,
10917:]
10914:1
10911:,
10908:0
10905:[
10899:t
10879:y
10876:,
10873:x
10853:f
10816:.
10814:R
10806:f
10801:.
10795:r
10791:x
10789:(
10787:f
10783:x
10781:{
10775:r
10767:m
10763:f
10733:X
10727:x
10707:0
10701:)
10698:x
10695:(
10688:f
10667:X
10647:f
10622:2
10614:x
10606:2
10603:m
10595:)
10592:x
10589:(
10586:f
10580:x
10570:m
10556:f
10533:)
10530:x
10524:y
10521:(
10516:T
10512:)
10508:x
10502:y
10499:(
10496:m
10490:)
10487:x
10481:y
10478:(
10475:)
10472:z
10469:(
10466:f
10461:2
10451:T
10447:)
10443:x
10437:y
10434:(
10414:)
10411:x
10405:y
10402:(
10399:)
10396:z
10393:(
10390:f
10385:2
10375:T
10371:)
10367:x
10361:y
10358:(
10353:2
10350:1
10345:+
10342:)
10339:x
10333:y
10330:(
10325:T
10321:)
10317:x
10314:(
10311:f
10305:+
10302:)
10299:x
10296:(
10293:f
10290:=
10287:)
10284:y
10281:(
10278:f
10258:}
10255:]
10252:1
10249:,
10246:0
10243:[
10237:t
10234::
10231:y
10228:)
10225:t
10219:1
10216:(
10213:+
10210:x
10207:t
10204:{
10198:z
10174:)
10171:x
10168:(
10165:f
10160:2
10132:0
10126:)
10123:x
10120:(
10113:f
10092:0
10089:=
10086:m
10066:m
10060:)
10057:x
10054:(
10047:f
10026:,
10023:)
10020:x
10017:(
10010:f
9989:)
9986:x
9983:(
9980:f
9975:2
9950:.
9947:x
9927:m
9907:)
9904:x
9901:(
9898:f
9893:2
9860:I
9857:m
9851:)
9848:x
9845:(
9842:f
9837:2
9788:f
9783:2
9758:I
9738:x
9718:I
9715:m
9709:)
9706:x
9703:(
9700:f
9695:2
9670:m
9650:f
9627:0
9621:m
9598:=
9595:m
9575:,
9572:0
9566:m
9542:2
9537:2
9529:y
9523:x
9517:)
9514:t
9508:1
9505:(
9502:t
9499:m
9494:2
9491:1
9483:)
9480:y
9477:(
9474:f
9471:)
9468:t
9462:1
9459:(
9456:+
9453:)
9450:x
9447:(
9444:f
9441:t
9435:)
9432:y
9429:)
9426:t
9420:1
9417:(
9414:+
9411:x
9408:t
9405:(
9402:f
9383:,
9380:]
9377:1
9374:,
9371:0
9368:[
9362:t
9342:y
9339:,
9336:x
9316:,
9313:m
9289:2
9284:2
9276:x
9270:y
9262:2
9259:m
9254:+
9251:)
9248:x
9242:y
9239:(
9234:T
9230:)
9226:x
9223:(
9220:f
9214:+
9211:)
9208:x
9205:(
9202:f
9196:)
9193:y
9190:(
9187:f
9120:,
9092:2
9084:y
9078:x
9072:m
9063:y
9057:x
9054:,
9051:)
9048:y
9045:(
9042:f
9033:)
9030:x
9027:(
9024:f
8996:2
8991:2
8983:y
8977:x
8971:m
8965:)
8962:y
8956:x
8953:(
8948:T
8944:)
8940:)
8937:y
8934:(
8931:f
8922:)
8919:x
8916:(
8913:f
8907:(
8887:y
8884:,
8881:x
8861:0
8855:m
8835:f
8812:)
8809:x
8806:(
8799:f
8778:0
8772:)
8767:n
8763:x
8759:(
8752:f
8728:n
8725:1
8719:=
8716:)
8711:n
8707:x
8703:(
8696:f
8675:)
8670:n
8666:x
8662:(
8642:f
8619:.
8616:x
8596:0
8590:m
8584:)
8581:x
8578:(
8571:f
8550:f
8528:x
8508:0
8502:)
8499:x
8496:(
8489:f
8468:f
8447:.
8444:x
8424:0
8418:)
8415:x
8412:(
8405:f
8384:f
8361:f
8326:b
8323:+
8320:a
8300:b
8297:,
8294:a
8274:X
8268:y
8265:,
8262:x
8242:)
8239:y
8236:(
8233:f
8230:b
8227:+
8224:)
8221:x
8218:(
8215:f
8212:a
8206:)
8203:y
8200:b
8197:+
8194:x
8191:a
8188:(
8185:f
8165:0
8159:)
8156:0
8153:(
8150:f
8129:R
8122:X
8119::
8116:f
8096:X
8072:}
8068:0
8062:t
8059:,
8056:)
8053:f
8050:(
8035:t
8032:x
8026::
8023:)
8020:t
8017:,
8014:x
8011:(
8007:{
7985:)
7979:t
7976:x
7970:(
7966:f
7963:t
7960:=
7957:)
7954:t
7951:,
7948:x
7945:(
7942:g
7922:f
7900:.
7888:)
7885:x
7882:(
7879:g
7859:C
7839:,
7836:x
7816:)
7813:y
7810:,
7807:x
7804:(
7801:f
7796:C
7790:y
7782:=
7779:)
7776:x
7773:(
7770:g
7750:)
7747:y
7744:,
7741:x
7738:(
7718:)
7715:y
7712:,
7709:x
7706:(
7703:f
7679:.
7674:n
7669:R
7659:g
7655:D
7632:m
7627:R
7619:b
7616:,
7611:n
7605:m
7600:R
7592:A
7572:)
7569:b
7566:+
7563:x
7560:A
7557:(
7554:f
7551:=
7548:)
7545:x
7542:(
7539:g
7517:m
7512:R
7502:f
7498:D
7477:f
7454:)
7451:)
7448:x
7445:(
7442:f
7439:(
7436:g
7433:=
7430:)
7427:x
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