2540:
1967:
2535:{\displaystyle {\begin{array}{lccccccccccccc}{\text{Minimize}}&f^{\top }x&+&g^{\top }y&+&p_{1}h_{1}^{\top }z_{1}&+&p_{2}h_{2}^{T}z_{2}&+&\cdots &+&p_{K}h_{K}^{\top }z_{K}&&\\{\text{subject to}}&Tx&+&Uy&&&&&&&&&=&r\\&&&V_{1}y&+&W_{1}z_{1}&&&&&&&=&s_{1}\\&&&V_{2}y&&&+&W_{2}z_{2}&&&&&=&s_{2}\\&&&\vdots &&&&&&\ddots &&&&\vdots \\&&&V_{K}y&&&&&&&+&W_{K}z_{K}&=&s_{K}\\&x&,&y&,&z_{1}&,&z_{2}&,&\ldots &,&z_{K}&\geq &0\\\end{array}}}
12557:, manually implementing explicit or implicit non-anticipativity to make sure the resulting model respects the structure of the information made available at each stage. An instance of an SP problem generated by a general modelling language tends to grow quite large (linearly in the number of scenarios), and its matrix loses the structure that is intrinsic to this class of problems, which could otherwise be exploited at solution time by specific decomposition algorithms. Extensions to modelling languages specifically designed for SP are starting to appear, see:
53:. This framework contrasts with deterministic optimization, in which all problem parameters are assumed to be known exactly. The goal of stochastic programming is to find a decision which both optimizes some criteria chosen by the decision maker, and appropriately accounts for the uncertainty of the problem parameters. Because many real-world decisions involve uncertainty, stochastic programming has found applications in a broad range of areas ranging from
1714:
11199:
3316:
10532:
12173:
1469:
12491:
11551:
1897:
mathematical program that can be used to compute the optimal first-stage decision, so these will exist for continuous probability distributions as well, when one can represent the second-stage cost in some closed form.) For example, to form the deterministic equivalent to the above stochastic linear program, we assign a probability
2680:
that is optimal using only a few scenarios provides more adaptable plans than one that assumes a single scenario only. In some cases such a claim could be verified by a simulation. In theory some measures of guarantee that an obtained solution solves the original problem with reasonable accuracy. Typically in applications only the
698:
10889:
3116:
511:
10245:
11913:
1709:{\displaystyle {\begin{array}{lccccccc}{\text{Minimize}}&f^{T}x&+&g^{T}y&+&h_{k}^{T}z_{k}&&\\{\text{subject to}}&Tx&+&Uy&&&=&r\\&&&V_{k}y&+&W_{k}z_{k}&=&s_{k}\\&x&,&y&,&z_{k}&\geq &0\end{array}}}
12227:
1002:
because the objective functions and the constraints are linear. Conceptually this is not essential and one can consider more general two-stage stochastic programs. For example, if the first-stage problem is integer, one could add integrality constraints to the first-stage problem so that the feasible
1896:
With a finite number of scenarios, two-stage stochastic linear programs can be modelled as large linear programming problems. This formulation is often called the deterministic equivalent linear program, or abbreviated to deterministic equivalent. (Strictly speaking a deterministic equivalent is any
2679:
In practice it might be possible to construct scenarios by eliciting experts' opinions on the future. The number of constructed scenarios should be relatively modest so that the obtained deterministic equivalent can be solved with reasonable computational effort. It is often claimed that a solution
912:
of the first-stage decision plus the expected cost of the (optimal) second-stage decision. We can view the second-stage problem simply as an optimization problem which describes our supposedly optimal behavior when the uncertain data is revealed, or we can consider its solution as a recourse action
11298:
106:
The basic idea of two-stage stochastic programming is that (optimal) decisions should be based on data available at the time the decisions are made and cannot depend on future observations. The two-stage formulation is widely used in stochastic programming. The general formulation of a two-stage
10052:. This can be done in various ways. For example, one can construct a particular scenario tree defining time evolution of the process. If at every stage the random return of each asset is allowed to have two continuations, independent of other assets, then the total number of scenarios is
1057:
could be inferred from historical data if one assumes that the distribution does not significantly change over the considered period of time. Also, the empirical distribution of the sample could be used as an approximation to the distribution of the future values of
553:
359:
4191:
11194:{\displaystyle {\begin{array}{lrclr}\max \limits _{x_{t}}&E})|\xi _{}]&\\{\text{subject to}}&W_{t+1}&=&\sum _{i=1}^{n}\xi _{i,t+1}x_{i,t}\\&\sum _{i=1}^{n}x_{i,t}&=&W_{t}\\&x_{t}&\geq &0\end{array}}}
3311:{\displaystyle {\begin{array}{rlrrr}{\hat {g}}_{N}(x)=&\min \limits _{x\in \mathbb {R} ^{n}}&c^{T}x+{\frac {1}{N}}\sum _{j=1}^{N}Q(x,\xi ^{j})&\\&{\text{subject to}}&Ax&=&b\\&&x&\geq &0\end{array}}}
10527:{\displaystyle {\begin{array}{lrclr}\max \limits _{x_{T-1}}&E}]&\\{\text{subject to}}&W_{T}&=&\sum _{i=1}^{n}\xi _{iT}x_{i,T-1}\\&\sum _{i=1}^{n}x_{i,T-1}&=&W_{T-1}\\&x_{T-1}&\geq &0\end{array}}}
12168:{\displaystyle {\begin{array}{lrclr}\max \limits _{x_{T-1}}&E&\\{\text{subject to}}&W_{T}&=&\sum _{i=1}^{n}\xi _{iT}x_{i,T-1}\\&\sum _{i=1}^{n}x_{i,T-1}&=&W_{T-1}\\&x_{T-1}&\geq &0\end{array}}}
12486:{\displaystyle {\begin{array}{lrclr}\max \limits _{x_{t}}&E&\\{\text{subject to}}&W_{t+1}&=&\sum _{i=1}^{n}\xi _{i,t+1}x_{i,t}\\&\sum _{i=1}^{n}x_{i,t}&=&W_{t}\\&x_{t}&\geq &0\end{array}}}
11546:{\displaystyle {\begin{array}{lrclr}\max \limits _{x_{0}}&E})]&\\{\text{subject to}}&W_{1}&=&\sum _{i=1}^{n}\xi _{i,1}x_{i0}\\&\sum _{i=1}^{n}x_{i0}&=&W_{0}\\&x_{0}&\geq &0\end{array}}}
6001:
1823:
involve only first-period variables and are the same in every scenario. The other constraints involve variables of later periods and differ in some respects from scenario to scenario, reflecting uncertainty about the future.
8175:
3107:
1887:
scenario in the second period with no uncertainty. In order to incorporate uncertainties in the second stage, one should assign probabilities to different scenarios and solve the corresponding deterministic equivalent.
2840:
A common approach to reduce the scenario set to a manageable size is by using Monte Carlo simulation. Suppose the total number of scenarios is very large or even infinite. Suppose further that we can generate a sample
2754:
independent random components, each of which has three possible realizations (for example, future realizations of each random parameters are classified as low, medium and high), then the total number of scenarios is
1403:
is a specific instance of the classical two-stage stochastic program. A stochastic LP is built from a collection of multi-period linear programs (LPs), each having the same structure but somewhat different data. The
8950:
invested in the respective assets. At that time the returns in the first period have been realized so it is reasonable to use this information in the rebalancing decision. Thus, the second-stage decisions, at time
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These questions are not independent. For example, the number of scenarios constructed will affect both the tractability of the deterministic equivalent and the quality of the obtained solutions.
208:
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is a useful tool in understanding decision making under uncertainty. The accumulation of capital stock under uncertainty is one example; often it is used by resource economists to analyze
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10006:. Optimization is performed over all implementable and feasible policies. To complete the problem description one also needs to define the probability distribution of the random process
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are the same in every scenario, however, because we must make a decision for the first period before we know which scenario will be realized. As a result, the constraints involving just
11692:
1357:
has an infinite (or very large) number of possible realizations the standard approach is then to represent this distribution by scenarios. This approach raises three questions, namely:
6190:
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10834:
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1179:
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693:{\displaystyle {\begin{array}{llr}\min \limits _{y\in \mathbb {R} ^{m}}&q(\xi )^{T}y&\\{\text{subject to}}&T(\xi )x+W(\xi )y=h(\xi )&\\&y\geq 0&\end{array}}}
9501:
9058:
8317:
wasps have shown the value of this modelling technique in explaining the evolution of behavioural decision making. These models are typically many-staged, rather than two-staged.
4909:
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and, moreover, the two-stage problem can be formulated as one large linear programming problem (this is called the deterministic equivalent of the original problem, see section
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5482:
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548:
506:{\displaystyle {\begin{array}{llr}\min \limits _{x\in \mathbb {R} ^{n}}&g(x)=c^{T}x+E_{\xi }&\\{\text{subject to}}&Ax=b&\\&x\geq 0&\end{array}}}
243:
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9002:
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993:
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8403:
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1972:
1922:
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1355:
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1076:
1055:
1029:
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934:
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7339:
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6434:
6284:
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6213:
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depending on the sample and therefore is random. Nevertheless, consistency results for SAA estimators can still be derived under some additional assumptions:
4489:. Therefore, it is natural to expect that the optimal value and optimal solutions of the SAA problem converge to their counterparts of the true problem as
248:
1380:, allows free access to many modern solvers. The structure of a deterministic equivalent is particularly amenable to apply decomposition methods, such as
7992:
2711:
has a practical value since almost always a "true" realization of the random data will be different from the set of constructed (generated) scenarios.
5903:
3008:
13009:
12672:
1230:
12938:
13129:
12574:
created to facilitate stochastic programming (includes keywords for parametric distributions, chance constraints and risk measures such as
13242:
13017:. MPS/SIAM Series on Optimization. Vol. 9. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). pp. xvi+436.
12680:. MPS/SIAM Series on Optimization. Vol. 9. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). pp. xvi+436.
110:
9751:
6750:
5677:
12600:
They both can generate SMPS instance level format, which conveys in a non-redundant form the structure of the problem to the solver.
6799:
5726:
12788:
13122:
7172:
2941:
819:
contains the data of the second-stage problem. In this formulation, at the first stage we have to make a "here-and-now" decision
12976:
12592:
specifically designed to express stochastic programs (includes syntax for chance constraints, integrated chance constraints and
4202:
12609:
9891:
Suppose the objective is to maximize the expected utility of this wealth at the last period, that is, to consider the problem
4710:
13093:
13022:
12763:
12685:
7887:
3723:
9591:
8704:
3501:
12922:"Using Polynomial Approximations to Solve Stochastic Dynamic Programming Problems: or A "Betty Crocker " Approach to SDP."
10126:
1474:
6030:
9216:
13145:
12865:
3327:
2844:
7380:
13196:
12957:
12909:
12889:
12571:
11588:, it may be hard to solve these dynamic programming equations. The situation simplifies dramatically if the process
13237:
12952:. Wiley-Interscience Series in Systems and Optimization. Chichester: John Wiley & Sons, Ltd. pp. xii+307.
7262:
1377:
9083:
8885:
8539:
7581:
7048:
3688:
8637:
4676:
11645:
2792:
of the number of scenarios makes model development using expert opinion very difficult even for reasonable size
11732:. In this case, the corresponding conditional expectations become unconditional expectations, and the function
8341:
8325:
8290:
6160:
5311:
3324:
method. The SAA problem is a function of the considered sample and in that sense is random. For a given sample
85:
79:
12921:
10759:
10009:
8813:
6902:
3924:
1138:
8347:
6475:
5626:
5219:
4600:
5001:
3383:
the SAA problem is of the same form as a two-stage stochastic linear programming problem with the scenarios
9448:
9007:
4862:
4186:{\displaystyle \min \limits _{x\in X}\{{\hat {g}}_{N}(x)=f(x)+{\frac {1}{N}}\sum _{j=1}^{N}Q(x,\xi ^{j})\}}
1962:. Then we can minimize the expected value of the objective, subject to the constraints from all scenarios:
740:
705:
13083:
13005:
12989:
12849:
12668:
11207:
9307:
9154:
6865:
6365:
5792:
5516:
5282:
5173:
4763:
4492:
4319:
3451:
13211:
12816:
12554:
7094:
2671:
need only be specified once, while the remaining constraints must be given separately for each scenario.
6708:
4914:
4796:
1227:. Then the expectation in the first-stage problem's objective function can be written as the summation:
1184:
1111:
To solve the two-stage stochastic problem numerically, one often needs to assume that the random vector
13191:
7507:
7438:
6974:
6291:
5818:
5442:
5099:
4414:
1373:
1037:
probability distribution. This would be justified in many situations. For example, the distribution of
38:
11855:
6127:
3619:
13036:
12699:
8352:
The following is an example from finance of multi-stage stochastic programming. Suppose that at time
4835:
4757:
be a sequence of (deterministic) real valued functions. The following two properties are equivalent:
1381:
27:
12499:
11778:
10842:
9544:
9369:
8428:
8247:
8182:
7627:
7344:
6668:
6520:
6439:
5590:
5255:
4636:
4546:
13067:
12730:
7961:
4047:, or it can be generated by Monte Carlo sampling techniques. Then we can formulate a corresponding
775:
50:
49:
is an optimization problem in which some or all problem parameters are uncertain, but follow known
12181:
11735:
11697:
9683:
9506:
8775:
3413:
1927:
13206:
12924:
University of
California, Davis, Department of Agricultural and Resource Economics Working Paper.
12644:
12232:
11918:
11303:
10894:
10250:
8302:
4597:
be the optimal value and the set of optimal solutions, respectively, of the true problem and let
3121:
90:
6635:
13161:
12624:
9445:
if it satisfies the model constraints with probability 1, i.e., the nonnegativity constraints
5046:
4952:
3886:
3763:
518:
213:
11618:
11591:
11564:
10612:
8980:
3386:
939:
13176:
13171:
12639:
11822:
10720:
10687:
10648:
10209:
10055:
9897:
8306:
7164:
2758:
1788:
1003:
set is discrete. Non-linear objectives and constraints could also be incorporated if needed.
95:
8607:
6948:
6607:
6391:
5542:
4345:
3860:
3834:
2947:
1860:
1830:
1437:
1407:
968:
885:
879:
becomes available, we optimize our behavior by solving an appropriate optimization problem.
13032:
12967:
12695:
10094:
9977:
9721:
9413:
8492:
8381:
7693:
7297:
6560:
6100:
4573:
4196:
2687:
2547:
1900:
1761:
8218:
7662:
7552:
7019:
6336:
6240:
5487:
5391:
5144:
4463:
4030:
3990:
3801:
3648:
2923:
2815:
2717:
1340:
1114:
1081:
1061:
1040:
1014:
862:
842:
8:
13216:
13186:
13166:
12983:
12593:
11269:
10540:
10088:
9951:
8954:
8859:
8355:
8298:
4669:
be the optimal value and the set of optimal solutions, respectively, of the SAA problem.
13088:. Springer Series in Operations Research and Financial Engineering. New York: Springer.
1758:
contain the first-period variables, whose values must be chosen immediately. The vector
916:
13156:
13054:
12856:. Handbooks in Operations Research and Management Science. Vol. 10. Philadelphia:
12717:
12579:
9871:
9287:
9063:
8519:
8472:
8408:
7487:
7324:
6587:
6419:
6269:
6218:
6198:
6008:
5877:
5857:
5570:
5420:
5369:
5349:
5199:
5075:
4981:
4526:
4275:
4010:
3970:
3668:
3599:
2903:
2795:
2737:
2654:
2634:
2614:
2594:
2574:
1741:
1721:
822:
35:
20:
4342:. Moreover, under mild additional conditions the convergence is uniform. We also have
13089:
13018:
13001:
12953:
12905:
12885:
12861:
12769:
12759:
12681:
12664:
10091:
equations, consider the above multistage problem backward in time. At the last stage
2835:
12977:
Optimization of
Stochastic Models. The Interface between Simulation and Optimization
9948:
This is a multistage stochastic programming problem, where stages are numbered from
9868:
which depends on the realization of the random process and the decisions up to time
558:
364:
356:
The classical two-stage linear stochastic programming problems can be formulated as
76:
Stochastic integer programming for problems in which some variables must be integers
12947:
12751:
6025:
of the SAA problem is estimated, then the corresponding SAA problem takes the form
1387:
How to measure quality of the obtained solution with respect to the "true" optimum.
859:, viewed as a random vector, is known. At the second stage, after a realization of
13114:
1011:
The formulation of the above two-stage problem assumes that the second-stage data
13028:
12963:
12691:
12614:
8170:{\displaystyle {\hat {\sigma }}^{2}(x):={\frac {1}{N-1}}\sum _{j=1}^{N}\left^{2}}
8425:
assets. Suppose further that we are allowed to rebalance our portfolio at times
12801:
Applications of stochastic programming are described at the following website,
1400:
58:
12755:
13231:
13201:
12996:. Handbooks in Operations Research and Management Science, Vol. 10, Elsevier.
12773:
12575:
12570:(Extended Mathematical Programming for Stochastic Programming) - a module of
12745:
5996:{\displaystyle \mathbb {D} (A,B):=\sup _{x\in A}\{\inf _{x'\in B}\|x-x'\|\}}
3102:{\displaystyle {\hat {q}}_{N}(x)={\frac {1}{N}}\sum _{j=1}^{N}Q(x,\xi ^{j})}
882:
At the first stage we optimize (minimize in the above formulation) the cost
82:
for dealing with constraints that must be satisfied with a given probability
1376:
can solve large linear/nonlinear problems. The NEOS Server, hosted at the
12553:
All discrete stochastic programming problems can be represented with any
6215:
of optimal solutions of the true problem is nonempty and is contained in
5366:
of optimal solutions of the true problem is nonempty and is contained in
3795:
is given by the optimal value of the corresponding second-stage problem.
42:
4543:
of the SAA problem is fixed, i.e., it is independent of the sample. Let
8314:
12986:. Stochastic Programming. Kluwer Academic Publishers, Dordrecht, 1995.
8604:
be the initial amounts invested in the n assets. We require that each
1785:
contains all of the variables for subsequent periods. The constraints
10239:
has been chosen. Therefore, one needs to solve the following problem
1331:
347:{\displaystyle \min _{y}\{q(y,\xi )\,|\,T(\xi )x+W(\xi )y=h(\xi )\}.}
16:
Framework for modeling optimization problems that involve uncertainty
12750:. Springer Series in Operations Research and Financial Engineering.
7271:
7228:
12857:
12835:
8294:
1891:
1098:, one could obtain a posteriori distribution by a Bayesian update.
2836:
Monte Carlo sampling and Sample
Average Approximation (SAA) Method
9858:{\displaystyle W_{t}=\sum _{i=1}^{n}\xi _{it}x_{i,t-1}(\xi _{}),}
6789:{\displaystyle {\hat {\vartheta }}_{N}\rightarrow \vartheta ^{*}}
5716:{\displaystyle {\hat {\vartheta }}_{N}\rightarrow \vartheta ^{*}}
54:
2812:. The situation becomes even worse if some random components of
12629:
8310:
6855:{\displaystyle \mathbb {D} (S^{*},{\hat {S}}_{N})\rightarrow 0}
5782:{\displaystyle \mathbb {D} (S^{*},{\hat {S}}_{N})\rightarrow 0}
8469:
but without injecting additional cash into it. At each period
1368:
How to solve the deterministic equivalent. Optimizers such as
12634:
12585:
12561:
8977:, are actually functions of realization of the random vector
8335:
7250:{\displaystyle {\sqrt {N}}{\xrightarrow {\mathcal {D}}}Y_{x}}
1369:
73:
Scenario-based methods including Sample
Average Approximation
69:
Several stochastic programming methods have been developed:
12999:
12662:
12589:
8882:, we can rebalance the portfolio by specifying the amounts
8489:
we make a decision about redistributing the current wealth
1323:{\displaystyle E=\sum \limits _{k=1}^{K}p_{k}Q(x,\xi _{k})}
9441:
of the decision process. It is said that such a policy is
4265:{\displaystyle {\frac {1}{N}}\sum _{j=1}^{N}Q(x,\xi ^{j})}
7987:
denotes the cdf of the standard normal distribution) and
4007:. This random sample can be viewed as historical data of
2944:(i.i.d sample). Given a sample, the expectation function
12902:
Models of adaptive behaviour: an approach based on state
4750:{\displaystyle {\hat {g}}_{N}:X\rightarrow \mathbb {R} }
3760:. In the framework of two-stage stochastic programming,
1464:
scenario, may be regarded as having the following form:
13011:
Lectures on stochastic programming: Modeling and theory
12920:
Howitt, R., Msangi, S., Reynaud, A and K. Knapp. 2002.
12674:
Lectures on stochastic programming: Modeling and theory
11556:
6894:
1332:§ Deterministic equivalent of a stochastic problem
936:
compensates for a possible inconsistency of the system
8332:
where the uncertainty enters in such as weather, etc.
7951:{\displaystyle z_{\alpha /2}:=\Phi ^{-1}(1-\alpha /2)}
3753:{\displaystyle Q:X\times \Xi \rightarrow \mathbb {R} }
3496:
Consider the following stochastic programming problem
13109:
13074:
Stein W. Wallace and
William T. Ziemba (eds.) (2005)
12802:
12502:
12230:
12184:
11916:
11858:
11825:
11781:
11738:
11700:
11648:
11621:
11594:
11567:
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11272:
11210:
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10762:
10723:
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10651:
10615:
10543:
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10212:
10129:
10097:
10058:
10012:
9980:
9954:
9900:
9874:
9754:
9724:
9686:
9670:{\displaystyle \sum _{i=1}^{n}x_{it}(\xi _{})=W_{t},}
9594:
9547:
9509:
9451:
9416:
9372:
9310:
9290:
9219:
9157:
9086:
9066:
9010:
8983:
8957:
8888:
8862:
8816:
8778:
8765:{\displaystyle \xi _{t}=(\xi _{1t},\dots ,\xi _{nt})}
8707:
8640:
8610:
8542:
8522:
8495:
8475:
8431:
8411:
8384:
8358:
8250:
8221:
8185:
7995:
7964:
7890:
7696:
7665:
7630:
7584:
7555:
7510:
7490:
7441:
7383:
7347:
7327:
7300:
7265:
7175:
7097:
7051:
7022:
6977:
6951:
6905:
6868:
6802:
6753:
6711:
6671:
6638:
6610:
6590:
6563:
6523:
6478:
6442:
6422:
6394:
6368:
6339:
6294:
6272:
6243:
6221:
6201:
6163:
6130:
6103:
6033:
6011:
5906:
5880:
5860:
5821:
5795:
5729:
5680:
5629:
5593:
5573:
5545:
5519:
5490:
5445:
5423:
5394:
5372:
5352:
5314:
5285:
5258:
5222:
5202:
5176:
5147:
5102:
5078:
5049:
5004:
4984:
4955:
4917:
4865:
4838:
4799:
4766:
4713:
4679:
4639:
4603:
4576:
4549:
4529:
4495:
4466:
4417:
4348:
4322:
4278:
4205:
4058:
4033:
4013:
3993:
3973:
3927:
3889:
3863:
3837:
3804:
3766:
3726:
3691:
3671:
3651:
3622:
3602:
3587:{\displaystyle \min \limits _{x\in X}\{g(x)=f(x)+E\}}
3504:
3454:
3416:
3389:
3330:
3119:
3111:
and consequently the first-stage problem is given by
3011:
2950:
2926:
2906:
2847:
2818:
2798:
2761:
2740:
2720:
2690:
2657:
2637:
2617:
2597:
2577:
2550:
1970:
1930:
1903:
1863:
1833:
1791:
1764:
1744:
1724:
1472:
1440:
1410:
1343:
1233:
1187:
1141:
1131:
has a finite number of possible realizations, called
1117:
1084:
1064:
1043:
1017:
971:
942:
919:
888:
865:
845:
825:
778:
743:
708:
556:
521:
362:
251:
216:
113:
10684:. The optimal value of the above problem depends on
10199:{\displaystyle \xi _{}=(\xi _{1},\dots ,\xi _{T-1})}
203:{\displaystyle \min _{x\in X}\{g(x)=f(x)+E_{\xi }\}}
13144:
6088:{\displaystyle \min _{x\in X_{N}}{\hat {g}}_{N}(x)}
12848:
12532:
12485:
12213:
12167:
11899:
11844:
11811:
11767:
11724:
11686:
11634:
11607:
11580:
11545:
11284:
11258:
11193:
10875:
10828:
10748:
10709:
10676:
10637:
10601:
10526:
10231:
10198:
10115:
10077:
10044:
9998:
9966:
9937:
9880:
9857:
9737:
9710:
9669:
9577:
9533:
9495:
9429:
9402:
9358:
9296:
9277:{\displaystyle \xi _{}=(\xi _{1},\dots ,\xi _{t})}
9276:
9205:
9143:
9072:
9052:
8996:
8969:
8942:
8874:
8848:
8802:
8764:
8690:
8626:
8596:
8528:
8508:
8481:
8461:
8417:
8397:
8370:
8269:
8236:
8207:
8169:
7979:
7950:
7874:
7680:
7651:
7616:
7570:
7541:
7496:
7472:
7424:
7369:
7333:
7313:
7282:
7249:
7155:
7083:
7037:
7008:
6963:
6937:
6880:
6854:
6788:
6730:
6697:
6657:
6622:
6596:
6576:
6549:
6506:
6464:
6428:
6406:
6380:
6354:
6325:
6278:
6258:
6227:
6207:
6184:
6145:
6116:
6087:
6017:
5995:
5886:
5866:
5844:
5807:
5781:
5715:
5657:
5615:
5579:
5557:
5531:
5505:
5476:
5429:
5409:
5378:
5358:
5335:
5297:
5271:
5244:
5208:
5188:
5162:
5133:
5084:
5064:
5035:
4990:
4970:
4939:
4903:
4851:
4824:
4785:
4749:
4699:
4661:
4625:
4589:
4562:
4535:
4507:
4481:
4448:
4403:
4334:
4308:
4264:
4185:
4039:
4019:
3999:
3979:
3959:
3910:
3875:
3849:
3819:
3787:
3752:
3712:
3677:
3665:is a random vector whose probability distribution
3657:
3637:
3608:
3586:
3480:
3440:
3402:
3375:
3310:
3101:
2995:
2932:
2912:
2892:
2824:
2804:
2780:
2746:
2726:
2703:
2663:
2643:
2623:
2603:
2583:
2563:
2534:
1954:
1916:
1879:
1849:
1815:
1777:
1750:
1730:
1708:
1456:
1426:
1349:
1322:
1219:
1173:
1123:
1090:
1070:
1049:
1023:
987:
957:
928:
904:
871:
851:
831:
811:
772:is the second-stage decision variable vector, and
764:
729:
692:
542:
505:
346:
237:
202:
3376:{\displaystyle \xi ^{1},\xi ^{2},\dots ,\xi ^{N}}
2893:{\displaystyle \xi ^{1},\xi ^{2},\dots ,\xi ^{N}}
550:is the optimal value of the second-stage problem
245:is the optimal value of the second-stage problem
13229:
9901:
7549:has approximately normal distribution with mean
7425:{\displaystyle {\mathcal {N}}(0,\sigma ^{2}(x))}
6035:
5950:
5931:
4518:
1892:Deterministic equivalent of a stochastic problem
1394:
253:
115:
12792:. MPS-SIAM Book Series on Optimization 5, 2005.
12786:Stein W. Wallace and William T. Ziemba (eds.).
4199:we have that, under some regularity conditions
101:
13130:
12743:
9284:the history of the random process up to time
8810:. This forms a vector-valued random process
8634:is nonnegative and that the balance equation
7283:{\displaystyle {\xrightarrow {\mathcal {D}}}}
839:before the realization of the uncertain data
737:is the first-stage decision variable vector,
9144:{\displaystyle x_{t}=(x_{1t},\dots ,x_{nt})}
8943:{\displaystyle x_{1}=(x_{11},\dots ,x_{n1})}
8597:{\displaystyle x_{0}=(x_{10},\dots ,x_{n0})}
8280:
7624:. This leads to the following (approximate)
7617:{\displaystyle {\frac {1}{N}}\sigma ^{2}(x)}
7084:{\displaystyle {\frac {1}{N}}\sigma ^{2}(x)}
5990:
5987:
5970:
5946:
4813:
4800:
4180:
4075:
3713:{\displaystyle \Xi \subset \mathbb {R} ^{d}}
3581:
3521:
2571:of later-period variables for each scenario
1857:two-period LP is equivalent to assuming the
1006:
338:
262:
197:
130:
107:stochastic programming problem is given by:
13081:
12945:
12814:
12744:Birge, John R.; Louveaux, François (2011).
8691:{\displaystyle \sum _{i=1}^{n}x_{i0}=W_{0}}
7163:is supposed to be finite. Moreover, by the
4700:{\displaystyle g:X\rightarrow \mathbb {R} }
13137:
13123:
13110:Stochastic Programming Community Home Page
12900:Houston, A. I & McNamara, J. M. 1999.
11687:{\displaystyle \xi _{1},\dots ,\xi _{t-1}}
11561:For a general distribution of the process
8336:Example: multistage portfolio optimization
8285:
5141:converges to the true problem's objective
4272:converges pointwise with probability 1 to
1362:
1101:
13082:King, Alan J.; Wallace, Stein W. (2012).
9585:, and the balance of wealth constraints,
6804:
6185:{\displaystyle C\subset \mathbb {R} ^{n}}
6172:
6133:
5908:
5823:
5731:
5336:{\displaystyle C\subset \mathbb {R} ^{n}}
5323:
4743:
4693:
3746:
3700:
3625:
3170:
752:
717:
574:
380:
289:
283:
13078:. MPS-SIAM Book Series on Optimization 5
12936:John R. Birge and François V. Louveaux.
12564:- supports the definition of SP problems
10829:{\displaystyle Q_{T-1}(W_{T-1},\xi _{})}
10045:{\displaystyle \xi _{1},\dots ,\xi _{T}}
8849:{\displaystyle \xi _{1},\dots ,\xi _{T}}
8320:
6938:{\displaystyle \xi ^{1},\dots ,\xi ^{N}}
6584:converges with probability 1 to a point
6157:Suppose that there exists a compact set
5308:Suppose that there exists a compact set
3960:{\displaystyle \xi ^{1},\dots ,\xi ^{N}}
3491:
2674:
1174:{\displaystyle \xi _{1},\dots ,\xi _{K}}
12946:Kall, Peter; Wallace, Stein W. (1994).
12882:Dynamic modeling in behavioral ecology.
10609:denotes the conditional expectation of
6507:{\displaystyle {\hat {S}}_{N}\subset C}
5658:{\displaystyle {\hat {S}}_{N}\subset C}
5245:{\displaystyle {\hat {\vartheta }}_{N}}
4626:{\displaystyle {\hat {\vartheta }}_{N}}
3448:, each taken with the same probability
2942:independent and identically distributed
19:For the context of control theory, see
13230:
13076:Applications of Stochastic Programming
12939:Introduction to Stochastic Programming
12789:Applications of Stochastic Programming
12747:Introduction to Stochastic Programming
12610:Chance-constrained portfolio selection
12548:
9213:of the available information given by
8215:. That is, the error of estimation of
5036:{\displaystyle {\hat {g}}_{N}(\cdot )}
3003:is approximated by the sample average
2940:. Usually the sample is assumed to be
13118:
9496:{\displaystyle x_{it}(\xi _{})\geq 0}
9053:{\displaystyle x_{1}=x_{1}(\xi _{1})}
4904:{\displaystyle {\hat {g}}_{N}(x_{N})}
1181:, with respective probability masses
1031:is modeled as a random vector with a
995:is the cost of this recourse action.
765:{\displaystyle y\in \mathbb {R} ^{m}}
730:{\displaystyle x\in \mathbb {R} ^{n}}
13085:Modeling with Stochastic Programming
12992:and Alexander Shapiro (eds.) (2003)
12880:Mangel, M. & Clark, C. W. 1988.
12818:A tutorial on Stochastic Programming
12815:Shapiro, Alexander; Philpott, Andy.
11907:is the optimal value of the problem
11557:Stagewise independent random process
11259:{\displaystyle Q_{t}(W_{t},\xi _{})}
9359:{\displaystyle x_{t}=x_{t}(\xi _{})}
9206:{\displaystyle x_{t}=x_{t}(\xi _{})}
7321:has a normal distribution with mean
6971:. Then the sample average estimator
6895:Asymptotics of the SAA optimal value
6881:{\displaystyle N\rightarrow \infty }
6381:{\displaystyle N\rightarrow \infty }
6005:In some situations the feasible set
5808:{\displaystyle N\rightarrow \infty }
5532:{\displaystyle N\rightarrow \infty }
5298:{\displaystyle N\rightarrow \infty }
5189:{\displaystyle N\rightarrow \infty }
5096:If the objective of the SAA problem
4786:{\displaystyle {\overline {x}}\in X}
4508:{\displaystyle N\rightarrow \infty }
4335:{\displaystyle N\rightarrow \infty }
3481:{\displaystyle p_{j}={\frac {1}{N}}}
998:The considered two-stage problem is
13243:Optimization algorithms and methods
12828:
11642:is (stochastically) independent of
10206:of the random process is known and
8179:is the sample variance estimate of
7156:{\displaystyle \sigma ^{2}(x):=Var}
6266:is finite valued and continuous on
5417:is finite valued and continuous on
5072:uniformly on any compact subset of
1265:
13:
12942:. Springer Verlag, New York, 1997.
12930:
11204:whose optimal value is denoted by
7965:
7913:
7386:
7273:
7230:
6875:
6731:{\displaystyle x_{N}\rightarrow x}
6375:
5802:
5526:
5292:
5183:
4940:{\displaystyle g({\overline {x}})}
4825:{\displaystyle \{x_{N}\}\subset X}
4502:
4329:
3987:realizations of the random vector
3739:
3692:
2920:replications of the random vector
2136:
2042:
2007:
1987:
1220:{\displaystyle p_{1},\dots ,p_{K}}
14:
13254:
13197:Infinite-dimensional optimization
13103:
12635:SAMPL algebraic modeling language
12543:
7542:{\displaystyle {\hat {g}}_{N}(x)}
7473:{\displaystyle {\hat {g}}_{N}(x)}
7009:{\displaystyle {\hat {g}}_{N}(x)}
6326:{\displaystyle {\hat {g}}_{N}(x)}
5845:{\displaystyle \mathbb {D} (A,B)}
5477:{\displaystyle {\hat {g}}_{N}(x)}
5134:{\displaystyle {\hat {g}}_{N}(x)}
4449:{\displaystyle {\hat {g}}_{N}(x)}
3320:This formulation is known as the
1106:
12803:Stochastic Programming Community
11900:{\displaystyle Q_{T-1}(W_{T-1})}
11615:is stagewise independent, i.e.,
8301:. Empirical tests of models of
6146:{\displaystyle \mathbb {R} ^{n}}
5196:, uniformly on the feasible set
3638:{\displaystyle \mathbb {R} ^{n}}
1434:two-period LP, representing the
1378:University of Wisconsin, Madison
1361:How to construct scenarios, see
13146:Major subfields of optimization
12914:
10883:, one should solve the problem
7045:, is unbiased and has variance
4852:{\displaystyle {\overline {x}}}
3616:is a nonempty closed subset of
2832:have continuous distributions.
1078:. If one has a prior model for
12894:
12874:
12842:
12836:"NEOS Server for Optimization"
12808:
12795:
12780:
12737:
12656:
12620:EMP for Stochastic Programming
12533:{\displaystyle t=T-2,\dots ,1}
12298:
12295:
12276:
12257:
12208:
12195:
11971:
11968:
11955:
11949:
11894:
11875:
11837:
11831:
11812:{\displaystyle t=1,\dots ,T-1}
11762:
11749:
11376:
11373:
11368:
11362:
11341:
11328:
11253:
11248:
11242:
11221:
11006:
11001:
10995:
10986:
10982:
10977:
10965:
10938:
10919:
10876:{\displaystyle t=T-2,\dots ,1}
10823:
10818:
10806:
10779:
10741:
10729:
10669:
10657:
10632:
10619:
10596:
10591:
10579:
10570:
10566:
10553:
10547:
10330:
10325:
10313:
10304:
10300:
10287:
10281:
10193:
10155:
10147:
10135:
9929:
9926:
9913:
9907:
9849:
9844:
9832:
9824:
9648:
9643:
9637:
9629:
9578:{\displaystyle t=0,\dots ,T-1}
9484:
9479:
9473:
9465:
9403:{\displaystyle t=0,\dots ,T-1}
9353:
9348:
9342:
9334:
9271:
9239:
9231:
9225:
9200:
9195:
9189:
9181:
9138:
9100:
9047:
9034:
8937:
8902:
8759:
8721:
8591:
8556:
8462:{\displaystyle t=1,\dots ,T-1}
8342:Intertemporal portfolio choice
8326:Stochastic dynamic programming
8291:Stochastic dynamic programming
8270:{\displaystyle O({\sqrt {N}})}
8264:
8254:
8231:
8225:
8208:{\displaystyle \sigma ^{2}(x)}
8202:
8196:
8153:
8134:
8094:
8075:
8021:
8015:
8003:
7974:
7968:
7945:
7925:
7856:
7850:
7844:
7811:
7805:
7793:
7772:
7766:
7760:
7727:
7721:
7709:
7675:
7669:
7652:{\displaystyle 100(1-\alpha )}
7646:
7634:
7611:
7605:
7565:
7559:
7536:
7530:
7518:
7484:distribution, i.e., for large
7467:
7461:
7449:
7419:
7416:
7410:
7391:
7370:{\displaystyle \sigma ^{2}(x)}
7364:
7358:
7220:
7217:
7211:
7193:
7183:
7150:
7147:
7135:
7129:
7114:
7108:
7078:
7072:
7032:
7026:
7003:
6997:
6985:
6872:
6846:
6843:
6831:
6808:
6773:
6761:
6722:
6698:{\displaystyle x_{N}\in X_{N}}
6550:{\displaystyle x_{N}\in X_{N}}
6486:
6465:{\displaystyle {\hat {S}}_{N}}
6450:
6372:
6349:
6343:
6320:
6314:
6302:
6253:
6247:
6082:
6076:
6064:
5924:
5912:
5839:
5827:
5799:
5773:
5770:
5758:
5735:
5700:
5688:
5637:
5616:{\displaystyle {\hat {S}}_{N}}
5601:
5523:
5500:
5494:
5471:
5465:
5453:
5404:
5398:
5289:
5272:{\displaystyle \vartheta ^{*}}
5230:
5180:
5157:
5151:
5128:
5122:
5110:
5059:
5053:
5030:
5024:
5012:
4965:
4959:
4934:
4921:
4898:
4885:
4873:
4739:
4721:
4689:
4662:{\displaystyle {\hat {S}}_{N}}
4647:
4611:
4563:{\displaystyle \vartheta ^{*}}
4499:
4476:
4470:
4443:
4437:
4425:
4398:
4392:
4383:
4380:
4374:
4362:
4352:
4326:
4303:
4300:
4288:
4282:
4259:
4240:
4177:
4158:
4118:
4112:
4103:
4097:
4085:
3921:Suppose that we have a sample
3905:
3893:
3857:. This implies that for every
3814:
3808:
3782:
3770:
3742:
3578:
3575:
3563:
3557:
3548:
3542:
3533:
3527:
3253:
3234:
3149:
3143:
3131:
3096:
3077:
3037:
3031:
3019:
2990:
2987:
2975:
2969:
2960:
2954:
1317:
1298:
1258:
1255:
1243:
1237:
806:
782:
667:
661:
649:
643:
631:
625:
598:
591:
537:
525:
456:
453:
441:
435:
403:
397:
335:
329:
317:
311:
299:
293:
285:
280:
268:
232:
220:
194:
191:
179:
173:
157:
151:
142:
136:
86:Stochastic dynamic programming
80:Chance constrained programming
1:
12904:. Cambridge University Press
12852:; Shapiro, Alexander (2003).
12650:
8244:is (stochastically) of order
7980:{\displaystyle \Phi (\cdot )}
4519:Consistency of SAA estimators
2591:. The first-period variables
1395:Stochastic linear programming
812:{\displaystyle \xi (q,T,W,h)}
12214:{\displaystyle Q_{t}(W_{t})}
11768:{\displaystyle Q_{t}(W_{t})}
11725:{\displaystyle t=2,\dots ,T}
9711:{\displaystyle t=1,\dots ,T}
9534:{\displaystyle i=1,\dots ,n}
8803:{\displaystyle t=1,\dots ,T}
8293:is frequently used to model
4929:
4844:
4772:
4049:sample average approximation
3441:{\displaystyle j=1,\dots ,N}
3322:Sample Average Approximation
1955:{\displaystyle k=1,\dots ,K}
1363:§ Scenario construction
102:Two-stage problem definition
7:
13212:Multiobjective optimization
12884:Princeton University Press
12603:
12555:algebraic modeling language
12236:
11922:
11307:
10898:
10254:
9437:being constant, defines an
8701:Consider the total returns
4060:
3506:
3158:
2544:We have a different vector
562:
368:
10:
13259:
13192:Combinatorial optimization
12980:. Kluwer, Dordrecht, 1996.
9304:. A sequence of functions
8348:Merton's portfolio problem
8345:
8339:
7659:% confidence interval for
6945:is i.i.d. and fix a point
6658:{\displaystyle x\in S^{*}}
6288:the sequence of functions
5439:the sequence of functions
1384:or scenario decomposition;
64:
18:
13152:
12756:10.1007/978-1-4614-0237-4
12588:- a set of extensions to
11292:, one solves the problem
8281:Applications and Examples
5065:{\displaystyle g(\cdot )}
4971:{\displaystyle g(\cdot )}
4523:Suppose the feasible set
3918:is finite almost surely.
3911:{\displaystyle Q(x,\xi )}
3788:{\displaystyle Q(x,\xi )}
1007:Distributional assumption
543:{\displaystyle Q(x,\xi )}
238:{\displaystyle Q(x,\xi )}
51:probability distributions
28:mathematical optimization
12221:is the optimal value of
11635:{\displaystyle \xi _{t}}
11608:{\displaystyle \xi _{t}}
11581:{\displaystyle \xi _{t}}
10638:{\displaystyle U(W_{T})}
8997:{\displaystyle \xi _{1}}
8378:we have initial capital
6665:there exists a sequence
3403:{\displaystyle \xi ^{j}}
958:{\displaystyle Tx\leq h}
61:to energy optimization.
13238:Stochastic optimization
13207:Constraint satisfaction
12645:Stochastic optimization
11845:{\displaystyle \xi _{}}
10749:{\displaystyle \xi _{}}
10710:{\displaystyle W_{T-1}}
10677:{\displaystyle \xi _{}}
10232:{\displaystyle x_{T-2}}
10078:{\displaystyle 2^{nT}.}
9938:{\displaystyle \max E.}
8286:Biological applications
7290:denotes convergence in
6362:with probability 1, as
5513:with probability 1, as
5170:with probability 1, as
2781:{\displaystyle K=3^{d}}
1816:{\displaystyle Tx+Uy=r}
1102:Scenario-based approach
91:Markov decision process
13182:Stochastic programming
13162:Fractional programming
12994:Stochastic Programming
12949:Stochastic programming
12854:Stochastic Programming
12625:Entropic value at risk
12534:
12487:
12420:
12356:
12215:
12169:
12084:
12023:
11901:
11846:
11813:
11769:
11726:
11688:
11636:
11609:
11582:
11547:
11483:
11428:
11286:
11260:
11195:
11128:
11064:
10877:
10830:
10750:
10711:
10678:
10639:
10603:
10528:
10443:
10382:
10233:
10200:
10117:
10079:
10046:
10000:
9968:
9939:
9882:
9859:
9788:
9739:
9712:
9671:
9615:
9579:
9535:
9497:
9431:
9404:
9360:
9298:
9278:
9207:
9145:
9074:
9054:
8998:
8971:
8944:
8876:
8850:
8804:
8766:
8692:
8661:
8628:
8627:{\displaystyle x_{i0}}
8598:
8530:
8510:
8483:
8463:
8419:
8399:
8372:
8271:
8238:
8209:
8171:
8130:
8065:
7981:
7952:
7876:
7682:
7653:
7618:
7572:
7543:
7498:
7474:
7426:
7371:
7335:
7315:
7284:
7251:
7157:
7085:
7039:
7010:
6965:
6964:{\displaystyle x\in X}
6939:
6882:
6862:with probability 1 as
6856:
6790:
6732:
6699:
6659:
6624:
6623:{\displaystyle x\in X}
6598:
6578:
6551:
6508:
6466:
6430:
6408:
6407:{\displaystyle x\in C}
6382:
6356:
6327:
6280:
6260:
6229:
6209:
6186:
6147:
6118:
6089:
6019:
5997:
5888:
5868:
5846:
5809:
5789:with probability 1 as
5783:
5717:
5659:
5617:
5581:
5559:
5558:{\displaystyle x\in C}
5533:
5507:
5478:
5431:
5411:
5380:
5360:
5337:
5299:
5279:with probability 1 as
5273:
5246:
5210:
5190:
5164:
5135:
5086:
5066:
5037:
4992:
4972:
4941:
4905:
4853:
4826:
4787:
4751:
4701:
4663:
4627:
4591:
4564:
4537:
4509:
4483:
4450:
4405:
4404:{\displaystyle E=g(x)}
4336:
4310:
4266:
4236:
4187:
4154:
4041:
4021:
4001:
3981:
3961:
3912:
3877:
3876:{\displaystyle x\in X}
3851:
3850:{\displaystyle x\in X}
3821:
3789:
3754:
3714:
3685:is supported on a set
3679:
3659:
3639:
3610:
3588:
3482:
3442:
3404:
3377:
3312:
3230:
3103:
3073:
2997:
2996:{\displaystyle q(x)=E}
2934:
2914:
2894:
2826:
2806:
2782:
2748:
2728:
2705:
2665:
2645:
2625:
2605:
2585:
2565:
2536:
1956:
1918:
1881:
1880:{\displaystyle k^{th}}
1851:
1850:{\displaystyle k^{th}}
1827:Note that solving the
1817:
1779:
1752:
1732:
1710:
1458:
1457:{\displaystyle k^{th}}
1428:
1427:{\displaystyle k^{th}}
1382:Benders' decomposition
1351:
1324:
1284:
1221:
1175:
1125:
1092:
1072:
1051:
1025:
989:
988:{\displaystyle q^{T}y}
959:
930:
906:
905:{\displaystyle c^{T}x}
873:
853:
833:
813:
766:
731:
694:
544:
507:
348:
239:
204:
41:problems that involve
32:stochastic programming
13177:Nonlinear programming
13172:Quadratic programming
12640:Scenario optimization
12535:
12488:
12400:
12336:
12216:
12170:
12064:
12003:
11902:
11847:
11814:
11770:
11727:
11689:
11637:
11610:
11583:
11548:
11463:
11408:
11287:
11261:
11196:
11108:
11044:
10878:
10839:Similarly, at stages
10831:
10751:
10712:
10679:
10640:
10604:
10529:
10423:
10362:
10234:
10201:
10118:
10116:{\displaystyle t=T-1}
10080:
10047:
10001:
9999:{\displaystyle t=T-1}
9969:
9940:
9883:
9860:
9768:
9740:
9738:{\displaystyle W_{t}}
9713:
9672:
9595:
9580:
9536:
9498:
9432:
9430:{\displaystyle x_{0}}
9405:
9361:
9299:
9279:
9208:
9146:
9075:
9060:. Similarly, at time
9055:
8999:
8972:
8945:
8877:
8851:
8805:
8767:
8693:
8641:
8629:
8599:
8531:
8511:
8509:{\displaystyle W_{t}}
8484:
8464:
8420:
8400:
8398:{\displaystyle W_{0}}
8373:
8321:Economic applications
8272:
8239:
8210:
8172:
8110:
8045:
7982:
7953:
7877:
7875:{\displaystyle \left}
7683:
7654:
7619:
7573:
7544:
7499:
7482:asymptotically normal
7475:
7427:
7372:
7336:
7316:
7314:{\displaystyle Y_{x}}
7285:
7252:
7165:central limit theorem
7158:
7086:
7040:
7011:
6966:
6940:
6883:
6857:
6791:
6733:
6700:
6660:
6625:
6599:
6579:
6577:{\displaystyle x_{N}}
6552:
6509:
6467:
6436:large enough the set
6431:
6409:
6383:
6357:
6328:
6281:
6261:
6230:
6210:
6187:
6148:
6119:
6117:{\displaystyle X_{N}}
6090:
6020:
5998:
5889:
5869:
5847:
5810:
5784:
5718:
5660:
5618:
5587:large enough the set
5582:
5560:
5534:
5508:
5479:
5432:
5412:
5381:
5361:
5338:
5300:
5274:
5247:
5211:
5191:
5165:
5136:
5087:
5067:
5038:
4993:
4973:
4942:
4906:
4854:
4827:
4788:
4752:
4702:
4664:
4628:
4592:
4590:{\displaystyle S^{*}}
4565:
4538:
4510:
4484:
4451:
4406:
4337:
4311:
4267:
4216:
4188:
4134:
4042:
4022:
4002:
3982:
3962:
3913:
3878:
3852:
3822:
3790:
3755:
3715:
3680:
3660:
3640:
3611:
3589:
3492:Statistical inference
3483:
3443:
3405:
3378:
3313:
3210:
3104:
3053:
2998:
2935:
2915:
2895:
2827:
2807:
2783:
2749:
2729:
2706:
2704:{\displaystyle x^{*}}
2675:Scenario construction
2666:
2646:
2626:
2606:
2586:
2566:
2564:{\displaystyle z_{k}}
2537:
1957:
1919:
1917:{\displaystyle p_{k}}
1882:
1852:
1818:
1780:
1778:{\displaystyle z_{k}}
1753:
1733:
1711:
1459:
1429:
1352:
1325:
1264:
1222:
1176:
1126:
1093:
1073:
1052:
1026:
990:
960:
931:
907:
874:
854:
834:
814:
767:
732:
695:
545:
508:
349:
240:
205:
96:Benders decomposition
13006:Ruszczyński, Andrzej
13000:Shapiro, Alexander;
12850:Ruszczyński, Andrzej
12669:Ruszczyński, Andrzej
12663:Shapiro, Alexander;
12500:
12228:
12182:
11914:
11856:
11823:
11779:
11736:
11698:
11646:
11619:
11592:
11565:
11299:
11270:
11266:. Finally, at stage
11208:
10890:
10843:
10760:
10721:
10688:
10649:
10613:
10541:
10246:
10210:
10127:
10095:
10056:
10010:
9978:
9952:
9898:
9872:
9752:
9722:
9684:
9592:
9545:
9507:
9449:
9439:implementable policy
9414:
9370:
9308:
9288:
9217:
9155:
9084:
9064:
9008:
8981:
8955:
8886:
8860:
8814:
8776:
8705:
8638:
8608:
8540:
8520:
8493:
8473:
8429:
8409:
8382:
8356:
8330:bioeconomic problems
8309:transitions such as
8248:
8237:{\displaystyle g(x)}
8219:
8183:
7993:
7962:
7888:
7694:
7681:{\displaystyle f(x)}
7663:
7628:
7582:
7571:{\displaystyle g(x)}
7553:
7508:
7488:
7439:
7381:
7345:
7325:
7298:
7263:
7173:
7095:
7049:
7038:{\displaystyle g(x)}
7020:
6975:
6949:
6903:
6866:
6800:
6751:
6709:
6669:
6636:
6608:
6588:
6561:
6521:
6476:
6440:
6420:
6392:
6366:
6355:{\displaystyle g(x)}
6337:
6292:
6270:
6259:{\displaystyle g(x)}
6241:
6219:
6199:
6161:
6128:
6101:
6031:
6009:
5904:
5878:
5858:
5819:
5793:
5727:
5678:
5627:
5591:
5571:
5543:
5517:
5506:{\displaystyle g(x)}
5488:
5443:
5421:
5410:{\displaystyle g(x)}
5392:
5370:
5350:
5312:
5283:
5256:
5220:
5200:
5174:
5163:{\displaystyle g(x)}
5145:
5100:
5076:
5047:
5002:
4982:
4953:
4915:
4863:
4836:
4797:
4764:
4711:
4677:
4637:
4601:
4574:
4547:
4527:
4493:
4482:{\displaystyle g(x)}
4464:
4415:
4346:
4320:
4276:
4203:
4197:Law of Large Numbers
4056:
4040:{\displaystyle \xi }
4031:
4011:
4000:{\displaystyle \xi }
3991:
3971:
3925:
3887:
3861:
3835:
3827:is well defined and
3820:{\displaystyle g(x)}
3802:
3764:
3724:
3689:
3669:
3658:{\displaystyle \xi }
3649:
3620:
3600:
3502:
3452:
3414:
3387:
3328:
3117:
3009:
2948:
2933:{\displaystyle \xi }
2924:
2904:
2845:
2825:{\displaystyle \xi }
2816:
2796:
2759:
2738:
2727:{\displaystyle \xi }
2718:
2688:
2655:
2635:
2615:
2595:
2575:
2548:
1968:
1928:
1901:
1861:
1831:
1789:
1762:
1742:
1722:
1470:
1438:
1408:
1350:{\displaystyle \xi }
1341:
1231:
1185:
1139:
1124:{\displaystyle \xi }
1115:
1091:{\displaystyle \xi }
1082:
1071:{\displaystyle \xi }
1062:
1050:{\displaystyle \xi }
1041:
1024:{\displaystyle \xi }
1015:
969:
940:
917:
886:
872:{\displaystyle \xi }
863:
852:{\displaystyle \xi }
843:
823:
776:
741:
706:
702:In such formulation
554:
519:
360:
249:
214:
111:
13217:Simulated annealing
13187:Robust optimization
13167:Integer programming
12990:Andrzej Ruszczynski
12594:Robust Optimization
12549:Modelling languages
11819:does not depend on
11285:{\displaystyle t=0}
10602:{\displaystyle E}]}
10089:dynamic programming
9967:{\displaystyle t=0}
8970:{\displaystyle t=1}
8875:{\displaystyle t=1}
8371:{\displaystyle t=0}
8299:behavioural ecology
7277:
7234:
6899:Suppose the sample
6738:with probability 1.
2140:
2088:
2046:
1538:
34:is a framework for
13157:Convex programming
13062:Unknown parameter
13002:Dentcheva, Darinka
12725:Unknown parameter
12665:Dentcheva, Darinka
12580:Expected shortfall
12530:
12483:
12481:
12251:
12211:
12165:
12163:
11943:
11897:
11842:
11809:
11765:
11722:
11684:
11632:
11605:
11578:
11543:
11541:
11322:
11282:
11256:
11191:
11189:
10913:
10873:
10826:
10746:
10707:
10674:
10635:
10599:
10524:
10522:
10275:
10229:
10196:
10113:
10087:In order to write
10075:
10042:
9996:
9964:
9935:
9878:
9855:
9735:
9708:
9667:
9575:
9531:
9493:
9427:
9400:
9356:
9294:
9274:
9203:
9141:
9070:
9050:
8994:
8967:
8940:
8872:
8846:
8800:
8762:
8688:
8624:
8594:
8526:
8506:
8479:
8459:
8415:
8395:
8368:
8313:and egg laying in
8297:in such fields as
8267:
8234:
8205:
8167:
7977:
7948:
7872:
7678:
7649:
7614:
7568:
7539:
7494:
7470:
7422:
7367:
7331:
7311:
7280:
7247:
7153:
7081:
7035:
7006:
6961:
6935:
6878:
6852:
6786:
6728:
6695:
6655:
6620:
6594:
6574:
6547:
6514:with probability 1
6504:
6462:
6426:
6404:
6378:
6352:
6323:
6276:
6256:
6225:
6205:
6182:
6143:
6114:
6085:
6056:
6015:
5993:
5969:
5945:
5884:
5864:
5842:
5805:
5779:
5713:
5665:with probability 1
5655:
5613:
5577:
5555:
5529:
5503:
5474:
5427:
5407:
5376:
5356:
5333:
5295:
5269:
5242:
5206:
5186:
5160:
5131:
5082:
5062:
5033:
4988:
4968:
4937:
4901:
4849:
4822:
4783:
4747:
4697:
4659:
4623:
4587:
4560:
4533:
4505:
4479:
4446:
4401:
4332:
4306:
4262:
4183:
4074:
4037:
4017:
3997:
3977:
3957:
3908:
3873:
3847:
3817:
3785:
3750:
3710:
3675:
3655:
3635:
3606:
3584:
3520:
3478:
3438:
3400:
3373:
3308:
3306:
3181:
3099:
2993:
2930:
2910:
2890:
2822:
2802:
2790:exponential growth
2778:
2744:
2724:
2701:
2661:
2641:
2621:
2601:
2581:
2561:
2532:
2530:
2126:
2074:
2032:
1952:
1914:
1877:
1847:
1813:
1775:
1748:
1728:
1706:
1704:
1524:
1454:
1424:
1347:
1320:
1217:
1171:
1121:
1088:
1068:
1047:
1021:
985:
955:
929:{\displaystyle Wy}
926:
902:
869:
849:
829:
809:
762:
727:
690:
688:
585:
540:
503:
501:
391:
344:
261:
235:
200:
129:
47:stochastic program
21:Stochastic control
13225:
13224:
13095:978-0-387-87816-4
13024:978-0-89871-687-0
12765:978-1-4614-0236-7
12687:978-0-89871-687-0
12309:
12235:
11982:
11921:
11387:
11306:
11017:
10897:
10341:
10253:
9881:{\displaystyle t}
9297:{\displaystyle t}
9073:{\displaystyle t}
8856:. At time period
8529:{\displaystyle n}
8482:{\displaystyle t}
8418:{\displaystyle n}
8311:fledging in birds
8262:
8108:
8043:
8006:
7865:
7864:
7847:
7796:
7781:
7780:
7763:
7712:
7593:
7521:
7497:{\displaystyle N}
7452:
7334:{\displaystyle 0}
7278:
7235:
7196:
7181:
7060:
6988:
6834:
6764:
6597:{\displaystyle x}
6489:
6453:
6429:{\displaystyle N}
6305:
6279:{\displaystyle C}
6228:{\displaystyle C}
6208:{\displaystyle S}
6067:
6034:
6018:{\displaystyle X}
5949:
5930:
5887:{\displaystyle B}
5867:{\displaystyle A}
5854:deviation of set
5761:
5691:
5640:
5604:
5580:{\displaystyle N}
5456:
5430:{\displaystyle C}
5379:{\displaystyle C}
5359:{\displaystyle S}
5233:
5209:{\displaystyle X}
5113:
5085:{\displaystyle X}
5015:
4991:{\displaystyle X}
4978:is continuous on
4932:
4876:
4847:
4793:and any sequence
4775:
4724:
4650:
4614:
4536:{\displaystyle X}
4428:
4365:
4309:{\displaystyle E}
4214:
4132:
4088:
4059:
4020:{\displaystyle N}
3980:{\displaystyle N}
3678:{\displaystyle P}
3609:{\displaystyle X}
3505:
3476:
3265:
3208:
3157:
3134:
3051:
3022:
2913:{\displaystyle N}
2805:{\displaystyle d}
2747:{\displaystyle d}
2684:optimal solution
2664:{\displaystyle y}
2644:{\displaystyle x}
2624:{\displaystyle y}
2604:{\displaystyle x}
2584:{\displaystyle k}
2160:
1978:
1924:to each scenario
1751:{\displaystyle y}
1731:{\displaystyle x}
1558:
1480:
832:{\displaystyle x}
618:
561:
467:
367:
252:
114:
13250:
13139:
13132:
13125:
13116:
13115:
13099:
13071:
13065:
13060:
13058:
13050:
13048:
13047:
13041:
13035:. Archived from
13016:
12971:
12925:
12918:
12912:
12898:
12892:
12878:
12872:
12871:
12846:
12840:
12839:
12832:
12826:
12825:
12823:
12812:
12806:
12799:
12793:
12784:
12778:
12777:
12741:
12735:
12734:
12728:
12723:
12721:
12713:
12711:
12710:
12704:
12698:. Archived from
12679:
12660:
12539:
12537:
12536:
12531:
12492:
12490:
12489:
12484:
12482:
12468:
12467:
12457:
12453:
12452:
12436:
12435:
12419:
12414:
12398:
12394:
12393:
12378:
12377:
12355:
12350:
12328:
12327:
12310:
12307:
12302:
12294:
12293:
12275:
12274:
12250:
12249:
12248:
12220:
12218:
12217:
12212:
12207:
12206:
12194:
12193:
12174:
12172:
12171:
12166:
12164:
12150:
12149:
12133:
12129:
12128:
12106:
12105:
12083:
12078:
12062:
12058:
12057:
12036:
12035:
12022:
12017:
11995:
11994:
11983:
11980:
11975:
11967:
11966:
11942:
11941:
11940:
11906:
11904:
11903:
11898:
11893:
11892:
11874:
11873:
11851:
11849:
11848:
11843:
11841:
11840:
11818:
11816:
11815:
11810:
11774:
11772:
11771:
11766:
11761:
11760:
11748:
11747:
11731:
11729:
11728:
11723:
11693:
11691:
11690:
11685:
11683:
11682:
11658:
11657:
11641:
11639:
11638:
11633:
11631:
11630:
11614:
11612:
11611:
11606:
11604:
11603:
11587:
11585:
11584:
11579:
11577:
11576:
11552:
11550:
11549:
11544:
11542:
11528:
11527:
11517:
11513:
11512:
11496:
11495:
11482:
11477:
11461:
11457:
11456:
11444:
11443:
11427:
11422:
11400:
11399:
11388:
11385:
11380:
11372:
11371:
11353:
11352:
11340:
11339:
11321:
11320:
11319:
11291:
11289:
11288:
11283:
11265:
11263:
11262:
11257:
11252:
11251:
11233:
11232:
11220:
11219:
11200:
11198:
11197:
11192:
11190:
11176:
11175:
11165:
11161:
11160:
11144:
11143:
11127:
11122:
11106:
11102:
11101:
11086:
11085:
11063:
11058:
11036:
11035:
11018:
11015:
11010:
11005:
11004:
10989:
10981:
10980:
10956:
10955:
10937:
10936:
10912:
10911:
10910:
10882:
10880:
10879:
10874:
10835:
10833:
10832:
10827:
10822:
10821:
10797:
10796:
10778:
10777:
10755:
10753:
10752:
10747:
10745:
10744:
10716:
10714:
10713:
10708:
10706:
10705:
10683:
10681:
10680:
10675:
10673:
10672:
10644:
10642:
10641:
10636:
10631:
10630:
10608:
10606:
10605:
10600:
10595:
10594:
10573:
10565:
10564:
10533:
10531:
10530:
10525:
10523:
10509:
10508:
10492:
10488:
10487:
10465:
10464:
10442:
10437:
10421:
10417:
10416:
10395:
10394:
10381:
10376:
10354:
10353:
10342:
10339:
10334:
10329:
10328:
10307:
10299:
10298:
10274:
10273:
10272:
10238:
10236:
10235:
10230:
10228:
10227:
10205:
10203:
10202:
10197:
10192:
10191:
10167:
10166:
10151:
10150:
10123:, a realization
10122:
10120:
10119:
10114:
10084:
10082:
10081:
10076:
10071:
10070:
10051:
10049:
10048:
10043:
10041:
10040:
10022:
10021:
10005:
10003:
10002:
9997:
9973:
9971:
9970:
9965:
9944:
9942:
9941:
9936:
9925:
9924:
9887:
9885:
9884:
9879:
9864:
9862:
9861:
9856:
9848:
9847:
9823:
9822:
9801:
9800:
9787:
9782:
9764:
9763:
9744:
9742:
9741:
9736:
9734:
9733:
9717:
9715:
9714:
9709:
9680:where in period
9676:
9674:
9673:
9668:
9663:
9662:
9647:
9646:
9628:
9627:
9614:
9609:
9584:
9582:
9581:
9576:
9540:
9538:
9537:
9532:
9502:
9500:
9499:
9494:
9483:
9482:
9464:
9463:
9436:
9434:
9433:
9428:
9426:
9425:
9409:
9407:
9406:
9401:
9365:
9363:
9362:
9357:
9352:
9351:
9333:
9332:
9320:
9319:
9303:
9301:
9300:
9295:
9283:
9281:
9280:
9275:
9270:
9269:
9251:
9250:
9235:
9234:
9212:
9210:
9209:
9204:
9199:
9198:
9180:
9179:
9167:
9166:
9150:
9148:
9147:
9142:
9137:
9136:
9115:
9114:
9096:
9095:
9079:
9077:
9076:
9071:
9059:
9057:
9056:
9051:
9046:
9045:
9033:
9032:
9020:
9019:
9003:
9001:
9000:
8995:
8993:
8992:
8976:
8974:
8973:
8968:
8949:
8947:
8946:
8941:
8936:
8935:
8914:
8913:
8898:
8897:
8881:
8879:
8878:
8873:
8855:
8853:
8852:
8847:
8845:
8844:
8826:
8825:
8809:
8807:
8806:
8801:
8772:for each period
8771:
8769:
8768:
8763:
8758:
8757:
8736:
8735:
8717:
8716:
8697:
8695:
8694:
8689:
8687:
8686:
8674:
8673:
8660:
8655:
8633:
8631:
8630:
8625:
8623:
8622:
8603:
8601:
8600:
8595:
8590:
8589:
8568:
8567:
8552:
8551:
8535:
8533:
8532:
8527:
8515:
8513:
8512:
8507:
8505:
8504:
8488:
8486:
8485:
8480:
8468:
8466:
8465:
8460:
8424:
8422:
8421:
8416:
8404:
8402:
8401:
8396:
8394:
8393:
8377:
8375:
8374:
8369:
8303:optimal foraging
8295:animal behaviour
8276:
8274:
8273:
8268:
8263:
8258:
8243:
8241:
8240:
8235:
8214:
8212:
8211:
8206:
8195:
8194:
8176:
8174:
8173:
8168:
8166:
8165:
8160:
8156:
8152:
8151:
8129:
8124:
8109:
8101:
8093:
8092:
8064:
8059:
8044:
8042:
8028:
8014:
8013:
8008:
8007:
7999:
7986:
7984:
7983:
7978:
7957:
7955:
7954:
7949:
7941:
7924:
7923:
7908:
7907:
7903:
7881:
7879:
7878:
7873:
7871:
7867:
7866:
7860:
7859:
7849:
7848:
7840:
7836:
7834:
7833:
7829:
7804:
7803:
7798:
7797:
7789:
7782:
7776:
7775:
7765:
7764:
7756:
7752:
7750:
7749:
7745:
7720:
7719:
7714:
7713:
7705:
7687:
7685:
7684:
7679:
7658:
7656:
7655:
7650:
7623:
7621:
7620:
7615:
7604:
7603:
7594:
7586:
7577:
7575:
7574:
7569:
7548:
7546:
7545:
7540:
7529:
7528:
7523:
7522:
7514:
7503:
7501:
7500:
7495:
7479:
7477:
7476:
7471:
7460:
7459:
7454:
7453:
7445:
7435:In other words,
7431:
7429:
7428:
7423:
7409:
7408:
7390:
7389:
7376:
7374:
7373:
7368:
7357:
7356:
7340:
7338:
7337:
7332:
7320:
7318:
7317:
7312:
7310:
7309:
7289:
7287:
7286:
7281:
7279:
7276:
7267:
7256:
7254:
7253:
7248:
7246:
7245:
7236:
7233:
7224:
7204:
7203:
7198:
7197:
7189:
7182:
7177:
7162:
7160:
7159:
7154:
7107:
7106:
7090:
7088:
7087:
7082:
7071:
7070:
7061:
7053:
7044:
7042:
7041:
7036:
7015:
7013:
7012:
7007:
6996:
6995:
6990:
6989:
6981:
6970:
6968:
6967:
6962:
6944:
6942:
6941:
6936:
6934:
6933:
6915:
6914:
6887:
6885:
6884:
6879:
6861:
6859:
6858:
6853:
6842:
6841:
6836:
6835:
6827:
6820:
6819:
6807:
6795:
6793:
6792:
6787:
6785:
6784:
6772:
6771:
6766:
6765:
6757:
6737:
6735:
6734:
6729:
6721:
6720:
6704:
6702:
6701:
6696:
6694:
6693:
6681:
6680:
6664:
6662:
6661:
6656:
6654:
6653:
6629:
6627:
6626:
6621:
6603:
6601:
6600:
6595:
6583:
6581:
6580:
6575:
6573:
6572:
6556:
6554:
6553:
6548:
6546:
6545:
6533:
6532:
6513:
6511:
6510:
6505:
6497:
6496:
6491:
6490:
6482:
6472:is nonempty and
6471:
6469:
6468:
6463:
6461:
6460:
6455:
6454:
6446:
6435:
6433:
6432:
6427:
6413:
6411:
6410:
6405:
6387:
6385:
6384:
6379:
6361:
6359:
6358:
6353:
6332:
6330:
6329:
6324:
6313:
6312:
6307:
6306:
6298:
6285:
6283:
6282:
6277:
6265:
6263:
6262:
6257:
6234:
6232:
6231:
6226:
6214:
6212:
6211:
6206:
6191:
6189:
6188:
6183:
6181:
6180:
6175:
6152:
6150:
6149:
6144:
6142:
6141:
6136:
6123:
6121:
6120:
6115:
6113:
6112:
6094:
6092:
6091:
6086:
6075:
6074:
6069:
6068:
6060:
6055:
6054:
6053:
6024:
6022:
6021:
6016:
6002:
6000:
5999:
5994:
5986:
5968:
5961:
5944:
5911:
5893:
5891:
5890:
5885:
5873:
5871:
5870:
5865:
5851:
5849:
5848:
5843:
5826:
5814:
5812:
5811:
5806:
5788:
5786:
5785:
5780:
5769:
5768:
5763:
5762:
5754:
5747:
5746:
5734:
5722:
5720:
5719:
5714:
5712:
5711:
5699:
5698:
5693:
5692:
5684:
5664:
5662:
5661:
5656:
5648:
5647:
5642:
5641:
5633:
5623:is nonempty and
5622:
5620:
5619:
5614:
5612:
5611:
5606:
5605:
5597:
5586:
5584:
5583:
5578:
5564:
5562:
5561:
5556:
5538:
5536:
5535:
5530:
5512:
5510:
5509:
5504:
5483:
5481:
5480:
5475:
5464:
5463:
5458:
5457:
5449:
5436:
5434:
5433:
5428:
5416:
5414:
5413:
5408:
5385:
5383:
5382:
5377:
5365:
5363:
5362:
5357:
5342:
5340:
5339:
5334:
5332:
5331:
5326:
5304:
5302:
5301:
5296:
5278:
5276:
5275:
5270:
5268:
5267:
5251:
5249:
5248:
5243:
5241:
5240:
5235:
5234:
5226:
5215:
5213:
5212:
5207:
5195:
5193:
5192:
5187:
5169:
5167:
5166:
5161:
5140:
5138:
5137:
5132:
5121:
5120:
5115:
5114:
5106:
5091:
5089:
5088:
5083:
5071:
5069:
5068:
5063:
5042:
5040:
5039:
5034:
5023:
5022:
5017:
5016:
5008:
4997:
4995:
4994:
4989:
4977:
4975:
4974:
4969:
4946:
4944:
4943:
4938:
4933:
4925:
4910:
4908:
4907:
4902:
4897:
4896:
4884:
4883:
4878:
4877:
4869:
4859:it follows that
4858:
4856:
4855:
4850:
4848:
4840:
4831:
4829:
4828:
4823:
4812:
4811:
4792:
4790:
4789:
4784:
4776:
4768:
4756:
4754:
4753:
4748:
4746:
4732:
4731:
4726:
4725:
4717:
4706:
4704:
4703:
4698:
4696:
4668:
4666:
4665:
4660:
4658:
4657:
4652:
4651:
4643:
4632:
4630:
4629:
4624:
4622:
4621:
4616:
4615:
4607:
4596:
4594:
4593:
4588:
4586:
4585:
4569:
4567:
4566:
4561:
4559:
4558:
4542:
4540:
4539:
4534:
4514:
4512:
4511:
4506:
4488:
4486:
4485:
4480:
4455:
4453:
4452:
4447:
4436:
4435:
4430:
4429:
4421:
4410:
4408:
4407:
4402:
4373:
4372:
4367:
4366:
4358:
4341:
4339:
4338:
4333:
4315:
4313:
4312:
4307:
4271:
4269:
4268:
4263:
4258:
4257:
4235:
4230:
4215:
4207:
4192:
4190:
4189:
4184:
4176:
4175:
4153:
4148:
4133:
4125:
4096:
4095:
4090:
4089:
4081:
4073:
4046:
4044:
4043:
4038:
4027:observations of
4026:
4024:
4023:
4018:
4006:
4004:
4003:
3998:
3986:
3984:
3983:
3978:
3966:
3964:
3963:
3958:
3956:
3955:
3937:
3936:
3917:
3915:
3914:
3909:
3882:
3880:
3879:
3874:
3856:
3854:
3853:
3848:
3826:
3824:
3823:
3818:
3794:
3792:
3791:
3786:
3759:
3757:
3756:
3751:
3749:
3719:
3717:
3716:
3711:
3709:
3708:
3703:
3684:
3682:
3681:
3676:
3664:
3662:
3661:
3656:
3644:
3642:
3641:
3636:
3634:
3633:
3628:
3615:
3613:
3612:
3607:
3593:
3591:
3590:
3585:
3519:
3487:
3485:
3484:
3479:
3477:
3469:
3464:
3463:
3447:
3445:
3444:
3439:
3409:
3407:
3406:
3401:
3399:
3398:
3382:
3380:
3379:
3374:
3372:
3371:
3353:
3352:
3340:
3339:
3317:
3315:
3314:
3309:
3307:
3289:
3288:
3266:
3263:
3260:
3257:
3252:
3251:
3229:
3224:
3209:
3201:
3193:
3192:
3180:
3179:
3178:
3173:
3142:
3141:
3136:
3135:
3127:
3108:
3106:
3105:
3100:
3095:
3094:
3072:
3067:
3052:
3044:
3030:
3029:
3024:
3023:
3015:
3002:
3000:
2999:
2994:
2939:
2937:
2936:
2931:
2919:
2917:
2916:
2911:
2899:
2897:
2896:
2891:
2889:
2888:
2870:
2869:
2857:
2856:
2831:
2829:
2828:
2823:
2811:
2809:
2808:
2803:
2787:
2785:
2784:
2779:
2777:
2776:
2753:
2751:
2750:
2745:
2733:
2731:
2730:
2725:
2710:
2708:
2707:
2702:
2700:
2699:
2670:
2668:
2667:
2662:
2650:
2648:
2647:
2642:
2630:
2628:
2627:
2622:
2610:
2608:
2607:
2602:
2590:
2588:
2587:
2582:
2570:
2568:
2567:
2562:
2560:
2559:
2541:
2539:
2538:
2533:
2531:
2517:
2516:
2490:
2489:
2473:
2472:
2442:
2438:
2437:
2421:
2420:
2411:
2410:
2395:
2394:
2393:
2392:
2391:
2390:
2385:
2384:
2374:
2373:
2372:
2364:
2363:
2362:
2356:
2355:
2354:
2353:
2352:
2346:
2345:
2344:
2340:
2339:
2324:
2323:
2322:
2321:
2319:
2318:
2309:
2308:
2293:
2292:
2287:
2286:
2276:
2275:
2274:
2270:
2269:
2254:
2253:
2252:
2251:
2250:
2249:
2247:
2246:
2237:
2236:
2217:
2216:
2206:
2205:
2204:
2191:
2190:
2189:
2188:
2187:
2186:
2185:
2184:
2161:
2158:
2153:
2152:
2150:
2149:
2139:
2134:
2125:
2124:
2098:
2097:
2087:
2082:
2073:
2072:
2056:
2055:
2045:
2040:
2031:
2030:
2011:
2010:
1991:
1990:
1979:
1976:
1961:
1959:
1958:
1953:
1923:
1921:
1920:
1915:
1913:
1912:
1886:
1884:
1883:
1878:
1876:
1875:
1856:
1854:
1853:
1848:
1846:
1845:
1822:
1820:
1819:
1814:
1784:
1782:
1781:
1776:
1774:
1773:
1757:
1755:
1754:
1749:
1737:
1735:
1734:
1729:
1715:
1713:
1712:
1707:
1705:
1691:
1690:
1660:
1656:
1655:
1639:
1638:
1629:
1628:
1609:
1608:
1598:
1597:
1596:
1583:
1582:
1559:
1556:
1551:
1550:
1548:
1547:
1537:
1532:
1513:
1512:
1493:
1492:
1481:
1478:
1463:
1461:
1460:
1455:
1453:
1452:
1433:
1431:
1430:
1425:
1423:
1422:
1356:
1354:
1353:
1348:
1329:
1327:
1326:
1321:
1316:
1315:
1294:
1293:
1283:
1278:
1226:
1224:
1223:
1218:
1216:
1215:
1197:
1196:
1180:
1178:
1177:
1172:
1170:
1169:
1151:
1150:
1130:
1128:
1127:
1122:
1097:
1095:
1094:
1089:
1077:
1075:
1074:
1069:
1056:
1054:
1053:
1048:
1030:
1028:
1027:
1022:
994:
992:
991:
986:
981:
980:
964:
962:
961:
956:
935:
933:
932:
927:
911:
909:
908:
903:
898:
897:
878:
876:
875:
870:
858:
856:
855:
850:
838:
836:
835:
830:
818:
816:
815:
810:
771:
769:
768:
763:
761:
760:
755:
736:
734:
733:
728:
726:
725:
720:
699:
697:
696:
691:
689:
686:
674:
671:
619:
616:
611:
606:
605:
584:
583:
582:
577:
549:
547:
546:
541:
512:
510:
509:
504:
502:
499:
487:
484:
468:
465:
460:
434:
433:
418:
417:
390:
389:
388:
383:
353:
351:
350:
345:
288:
260:
244:
242:
241:
236:
209:
207:
206:
201:
172:
171:
128:
26:In the field of
13258:
13257:
13253:
13252:
13251:
13249:
13248:
13247:
13228:
13227:
13226:
13221:
13148:
13143:
13106:
13096:
13063:
13061:
13052:
13051:
13045:
13043:
13039:
13025:
13014:
12974:G. Ch. Pflug:
12960:
12933:
12931:Further reading
12928:
12919:
12915:
12899:
12895:
12879:
12875:
12868:
12860:. p. 700.
12847:
12843:
12834:
12833:
12829:
12821:
12813:
12809:
12800:
12796:
12785:
12781:
12766:
12742:
12738:
12726:
12724:
12715:
12714:
12708:
12706:
12702:
12688:
12677:
12661:
12657:
12653:
12615:Correlation gap
12606:
12551:
12546:
12501:
12498:
12497:
12480:
12479:
12474:
12469:
12463:
12459:
12455:
12454:
12448:
12444:
12442:
12437:
12425:
12421:
12415:
12404:
12396:
12395:
12383:
12379:
12361:
12357:
12351:
12340:
12334:
12329:
12317:
12313:
12311:
12306:
12303:
12301:
12283:
12279:
12264:
12260:
12252:
12244:
12240:
12239:
12231:
12229:
12226:
12225:
12202:
12198:
12189:
12185:
12183:
12180:
12179:
12162:
12161:
12156:
12151:
12139:
12135:
12131:
12130:
12118:
12114:
12112:
12107:
12089:
12085:
12079:
12068:
12060:
12059:
12041:
12037:
12028:
12024:
12018:
12007:
12001:
11996:
11990:
11986:
11984:
11979:
11976:
11974:
11962:
11958:
11944:
11930:
11926:
11925:
11917:
11915:
11912:
11911:
11882:
11878:
11863:
11859:
11857:
11854:
11853:
11830:
11826:
11824:
11821:
11820:
11780:
11777:
11776:
11756:
11752:
11743:
11739:
11737:
11734:
11733:
11699:
11696:
11695:
11672:
11668:
11653:
11649:
11647:
11644:
11643:
11626:
11622:
11620:
11617:
11616:
11599:
11595:
11593:
11590:
11589:
11572:
11568:
11566:
11563:
11562:
11559:
11540:
11539:
11534:
11529:
11523:
11519:
11515:
11514:
11508:
11504:
11502:
11497:
11488:
11484:
11478:
11467:
11459:
11458:
11449:
11445:
11433:
11429:
11423:
11412:
11406:
11401:
11395:
11391:
11389:
11384:
11381:
11379:
11361:
11357:
11348:
11344:
11335:
11331:
11323:
11315:
11311:
11310:
11302:
11300:
11297:
11296:
11271:
11268:
11267:
11241:
11237:
11228:
11224:
11215:
11211:
11209:
11206:
11205:
11188:
11187:
11182:
11177:
11171:
11167:
11163:
11162:
11156:
11152:
11150:
11145:
11133:
11129:
11123:
11112:
11104:
11103:
11091:
11087:
11069:
11065:
11059:
11048:
11042:
11037:
11025:
11021:
11019:
11014:
11011:
11009:
10994:
10990:
10985:
10964:
10960:
10945:
10941:
10926:
10922:
10914:
10906:
10902:
10901:
10893:
10891:
10888:
10887:
10844:
10841:
10840:
10805:
10801:
10786:
10782:
10767:
10763:
10761:
10758:
10757:
10756:and is denoted
10728:
10724:
10722:
10719:
10718:
10695:
10691:
10689:
10686:
10685:
10656:
10652:
10650:
10647:
10646:
10626:
10622:
10614:
10611:
10610:
10578:
10574:
10569:
10560:
10556:
10542:
10539:
10538:
10521:
10520:
10515:
10510:
10498:
10494:
10490:
10489:
10477:
10473:
10471:
10466:
10448:
10444:
10438:
10427:
10419:
10418:
10400:
10396:
10387:
10383:
10377:
10366:
10360:
10355:
10349:
10345:
10343:
10338:
10335:
10333:
10312:
10308:
10303:
10294:
10290:
10276:
10262:
10258:
10257:
10249:
10247:
10244:
10243:
10217:
10213:
10211:
10208:
10207:
10181:
10177:
10162:
10158:
10134:
10130:
10128:
10125:
10124:
10096:
10093:
10092:
10063:
10059:
10057:
10054:
10053:
10036:
10032:
10017:
10013:
10011:
10008:
10007:
9979:
9976:
9975:
9953:
9950:
9949:
9920:
9916:
9899:
9896:
9895:
9873:
9870:
9869:
9831:
9827:
9806:
9802:
9793:
9789:
9783:
9772:
9759:
9755:
9753:
9750:
9749:
9729:
9725:
9723:
9720:
9719:
9685:
9682:
9681:
9658:
9654:
9636:
9632:
9620:
9616:
9610:
9599:
9593:
9590:
9589:
9546:
9543:
9542:
9508:
9505:
9504:
9472:
9468:
9456:
9452:
9450:
9447:
9446:
9421:
9417:
9415:
9412:
9411:
9371:
9368:
9367:
9341:
9337:
9328:
9324:
9315:
9311:
9309:
9306:
9305:
9289:
9286:
9285:
9265:
9261:
9246:
9242:
9224:
9220:
9218:
9215:
9214:
9188:
9184:
9175:
9171:
9162:
9158:
9156:
9153:
9152:
9129:
9125:
9107:
9103:
9091:
9087:
9085:
9082:
9081:
9065:
9062:
9061:
9041:
9037:
9028:
9024:
9015:
9011:
9009:
9006:
9005:
8988:
8984:
8982:
8979:
8978:
8956:
8953:
8952:
8928:
8924:
8909:
8905:
8893:
8889:
8887:
8884:
8883:
8861:
8858:
8857:
8840:
8836:
8821:
8817:
8815:
8812:
8811:
8777:
8774:
8773:
8750:
8746:
8728:
8724:
8712:
8708:
8706:
8703:
8702:
8682:
8678:
8666:
8662:
8656:
8645:
8639:
8636:
8635:
8615:
8611:
8609:
8606:
8605:
8582:
8578:
8563:
8559:
8547:
8543:
8541:
8538:
8537:
8521:
8518:
8517:
8500:
8496:
8494:
8491:
8490:
8474:
8471:
8470:
8430:
8427:
8426:
8410:
8407:
8406:
8389:
8385:
8383:
8380:
8379:
8357:
8354:
8353:
8350:
8344:
8338:
8323:
8288:
8283:
8257:
8249:
8246:
8245:
8220:
8217:
8216:
8190:
8186:
8184:
8181:
8180:
8177:
8161:
8147:
8143:
8125:
8114:
8100:
8088:
8084:
8071:
8067:
8066:
8060:
8049:
8032:
8027:
8009:
7998:
7997:
7996:
7994:
7991:
7990:
7963:
7960:
7959:
7937:
7916:
7912:
7899:
7895:
7891:
7889:
7886:
7885:
7882:
7839:
7838:
7837:
7835:
7825:
7821:
7817:
7799:
7788:
7787:
7786:
7755:
7754:
7753:
7751:
7741:
7737:
7733:
7715:
7704:
7703:
7702:
7701:
7697:
7695:
7692:
7691:
7664:
7661:
7660:
7629:
7626:
7625:
7599:
7595:
7585:
7583:
7580:
7579:
7554:
7551:
7550:
7524:
7513:
7512:
7511:
7509:
7506:
7505:
7489:
7486:
7485:
7455:
7444:
7443:
7442:
7440:
7437:
7436:
7404:
7400:
7385:
7384:
7382:
7379:
7378:
7352:
7348:
7346:
7343:
7342:
7326:
7323:
7322:
7305:
7301:
7299:
7296:
7295:
7272:
7266:
7264:
7261:
7260:
7257:
7241:
7237:
7229:
7223:
7199:
7188:
7187:
7186:
7176:
7174:
7171:
7170:
7102:
7098:
7096:
7093:
7092:
7066:
7062:
7052:
7050:
7047:
7046:
7021:
7018:
7017:
6991:
6980:
6979:
6978:
6976:
6973:
6972:
6950:
6947:
6946:
6929:
6925:
6910:
6906:
6904:
6901:
6900:
6897:
6867:
6864:
6863:
6837:
6826:
6825:
6824:
6815:
6811:
6803:
6801:
6798:
6797:
6780:
6776:
6767:
6756:
6755:
6754:
6752:
6749:
6748:
6716:
6712:
6710:
6707:
6706:
6689:
6685:
6676:
6672:
6670:
6667:
6666:
6649:
6645:
6637:
6634:
6633:
6632:for some point
6609:
6606:
6605:
6589:
6586:
6585:
6568:
6564:
6562:
6559:
6558:
6541:
6537:
6528:
6524:
6522:
6519:
6518:
6492:
6481:
6480:
6479:
6477:
6474:
6473:
6456:
6445:
6444:
6443:
6441:
6438:
6437:
6421:
6418:
6417:
6393:
6390:
6389:
6388:, uniformly in
6367:
6364:
6363:
6338:
6335:
6334:
6308:
6297:
6296:
6295:
6293:
6290:
6289:
6271:
6268:
6267:
6242:
6239:
6238:
6220:
6217:
6216:
6200:
6197:
6196:
6176:
6171:
6170:
6162:
6159:
6158:
6137:
6132:
6131:
6129:
6126:
6125:
6124:is a subset of
6108:
6104:
6102:
6099:
6098:
6095:
6070:
6059:
6058:
6057:
6049:
6045:
6038:
6032:
6029:
6028:
6010:
6007:
6006:
6003:
5979:
5954:
5953:
5934:
5907:
5905:
5902:
5901:
5879:
5876:
5875:
5859:
5856:
5855:
5822:
5820:
5817:
5816:
5794:
5791:
5790:
5764:
5753:
5752:
5751:
5742:
5738:
5730:
5728:
5725:
5724:
5707:
5703:
5694:
5683:
5682:
5681:
5679:
5676:
5675:
5643:
5632:
5631:
5630:
5628:
5625:
5624:
5607:
5596:
5595:
5594:
5592:
5589:
5588:
5572:
5569:
5568:
5544:
5541:
5540:
5539:, uniformly in
5518:
5515:
5514:
5489:
5486:
5485:
5459:
5448:
5447:
5446:
5444:
5441:
5440:
5422:
5419:
5418:
5393:
5390:
5389:
5371:
5368:
5367:
5351:
5348:
5347:
5327:
5322:
5321:
5313:
5310:
5309:
5284:
5281:
5280:
5263:
5259:
5257:
5254:
5253:
5236:
5225:
5224:
5223:
5221:
5218:
5217:
5201:
5198:
5197:
5175:
5172:
5171:
5146:
5143:
5142:
5116:
5105:
5104:
5103:
5101:
5098:
5097:
5077:
5074:
5073:
5048:
5045:
5044:
5018:
5007:
5006:
5005:
5003:
5000:
4999:
4983:
4980:
4979:
4954:
4951:
4950:
4924:
4916:
4913:
4912:
4892:
4888:
4879:
4868:
4867:
4866:
4864:
4861:
4860:
4839:
4837:
4834:
4833:
4807:
4803:
4798:
4795:
4794:
4767:
4765:
4762:
4761:
4742:
4727:
4716:
4715:
4714:
4712:
4709:
4708:
4692:
4678:
4675:
4674:
4653:
4642:
4641:
4640:
4638:
4635:
4634:
4617:
4606:
4605:
4604:
4602:
4599:
4598:
4581:
4577:
4575:
4572:
4571:
4554:
4550:
4548:
4545:
4544:
4528:
4525:
4524:
4521:
4494:
4491:
4490:
4465:
4462:
4461:
4431:
4420:
4419:
4418:
4416:
4413:
4412:
4368:
4357:
4356:
4355:
4347:
4344:
4343:
4321:
4318:
4317:
4277:
4274:
4273:
4253:
4249:
4231:
4220:
4206:
4204:
4201:
4200:
4193:
4171:
4167:
4149:
4138:
4124:
4091:
4080:
4079:
4078:
4063:
4057:
4054:
4053:
4032:
4029:
4028:
4012:
4009:
4008:
3992:
3989:
3988:
3972:
3969:
3968:
3951:
3947:
3932:
3928:
3926:
3923:
3922:
3888:
3885:
3884:
3862:
3859:
3858:
3836:
3833:
3832:
3803:
3800:
3799:
3765:
3762:
3761:
3745:
3725:
3722:
3721:
3704:
3699:
3698:
3690:
3687:
3686:
3670:
3667:
3666:
3650:
3647:
3646:
3629:
3624:
3623:
3621:
3618:
3617:
3601:
3598:
3597:
3594:
3509:
3503:
3500:
3499:
3494:
3468:
3459:
3455:
3453:
3450:
3449:
3415:
3412:
3411:
3394:
3390:
3388:
3385:
3384:
3367:
3363:
3348:
3344:
3335:
3331:
3329:
3326:
3325:
3305:
3304:
3299:
3294:
3286:
3285:
3280:
3275:
3267:
3262:
3258:
3256:
3247:
3243:
3225:
3214:
3200:
3188:
3184:
3182:
3174:
3169:
3168:
3161:
3155:
3137:
3126:
3125:
3124:
3120:
3118:
3115:
3114:
3090:
3086:
3068:
3057:
3043:
3025:
3014:
3013:
3012:
3010:
3007:
3006:
2949:
2946:
2945:
2925:
2922:
2921:
2905:
2902:
2901:
2884:
2880:
2865:
2861:
2852:
2848:
2846:
2843:
2842:
2838:
2817:
2814:
2813:
2797:
2794:
2793:
2772:
2768:
2760:
2757:
2756:
2739:
2736:
2735:
2719:
2716:
2715:
2695:
2691:
2689:
2686:
2685:
2677:
2656:
2653:
2652:
2636:
2633:
2632:
2616:
2613:
2612:
2596:
2593:
2592:
2576:
2573:
2572:
2555:
2551:
2549:
2546:
2545:
2529:
2528:
2523:
2518:
2512:
2508:
2506:
2501:
2496:
2491:
2485:
2481:
2479:
2474:
2468:
2464:
2462:
2457:
2452:
2447:
2440:
2439:
2433:
2429:
2427:
2422:
2416:
2412:
2406:
2402:
2400:
2389:
2380:
2376:
2370:
2369:
2361:
2351:
2342:
2341:
2335:
2331:
2329:
2320:
2314:
2310:
2304:
2300:
2298:
2291:
2282:
2278:
2272:
2271:
2265:
2261:
2259:
2248:
2242:
2238:
2232:
2228:
2226:
2221:
2212:
2208:
2202:
2201:
2196:
2183:
2175:
2170:
2162:
2157:
2154:
2151:
2145:
2141:
2135:
2130:
2120:
2116:
2114:
2109:
2104:
2099:
2093:
2089:
2083:
2078:
2068:
2064:
2062:
2057:
2051:
2047:
2041:
2036:
2026:
2022:
2020:
2015:
2006:
2002:
2000:
1995:
1986:
1982:
1980:
1975:
1971:
1969:
1966:
1965:
1929:
1926:
1925:
1908:
1904:
1902:
1899:
1898:
1894:
1868:
1864:
1862:
1859:
1858:
1838:
1834:
1832:
1829:
1828:
1790:
1787:
1786:
1769:
1765:
1763:
1760:
1759:
1743:
1740:
1739:
1723:
1720:
1719:
1703:
1702:
1697:
1692:
1686:
1682:
1680:
1675:
1670:
1665:
1658:
1657:
1651:
1647:
1645:
1640:
1634:
1630:
1624:
1620:
1618:
1613:
1604:
1600:
1594:
1593:
1588:
1581:
1573:
1568:
1560:
1555:
1552:
1549:
1543:
1539:
1533:
1528:
1522:
1517:
1508:
1504:
1502:
1497:
1488:
1484:
1482:
1477:
1473:
1471:
1468:
1467:
1445:
1441:
1439:
1436:
1435:
1415:
1411:
1409:
1406:
1405:
1397:
1342:
1339:
1338:
1311:
1307:
1289:
1285:
1279:
1268:
1232:
1229:
1228:
1211:
1207:
1192:
1188:
1186:
1183:
1182:
1165:
1161:
1146:
1142:
1140:
1137:
1136:
1116:
1113:
1112:
1109:
1104:
1083:
1080:
1079:
1063:
1060:
1059:
1042:
1039:
1038:
1016:
1013:
1012:
1009:
976:
972:
970:
967:
966:
941:
938:
937:
918:
915:
914:
913:where the term
893:
889:
887:
884:
883:
864:
861:
860:
844:
841:
840:
824:
821:
820:
777:
774:
773:
756:
751:
750:
742:
739:
738:
721:
716:
715:
707:
704:
703:
687:
685:
672:
670:
620:
615:
612:
610:
601:
597:
586:
578:
573:
572:
565:
557:
555:
552:
551:
520:
517:
516:
500:
498:
485:
483:
469:
464:
461:
459:
429:
425:
413:
409:
392:
384:
379:
378:
371:
363:
361:
358:
357:
284:
256:
250:
247:
246:
215:
212:
211:
167:
163:
118:
112:
109:
108:
104:
67:
24:
17:
12:
11:
5:
13256:
13246:
13245:
13240:
13223:
13222:
13220:
13219:
13214:
13209:
13204:
13202:Metaheuristics
13199:
13194:
13189:
13184:
13179:
13174:
13169:
13164:
13159:
13153:
13150:
13149:
13142:
13141:
13134:
13127:
13119:
13113:
13112:
13105:
13104:External links
13102:
13101:
13100:
13094:
13079:
13072:
13023:
12997:
12987:
12984:András Prékopa
12981:
12972:
12958:
12943:
12932:
12929:
12927:
12926:
12913:
12893:
12873:
12867:978-0444508546
12866:
12841:
12827:
12807:
12794:
12779:
12764:
12736:
12686:
12654:
12652:
12649:
12648:
12647:
12642:
12637:
12632:
12627:
12622:
12617:
12612:
12605:
12602:
12598:
12597:
12583:
12565:
12550:
12547:
12545:
12544:Software tools
12542:
12529:
12526:
12523:
12520:
12517:
12514:
12511:
12508:
12505:
12494:
12493:
12478:
12475:
12473:
12470:
12466:
12462:
12458:
12456:
12451:
12447:
12443:
12441:
12438:
12434:
12431:
12428:
12424:
12418:
12413:
12410:
12407:
12403:
12399:
12397:
12392:
12389:
12386:
12382:
12376:
12373:
12370:
12367:
12364:
12360:
12354:
12349:
12346:
12343:
12339:
12335:
12333:
12330:
12326:
12323:
12320:
12316:
12312:
12305:
12304:
12300:
12297:
12292:
12289:
12286:
12282:
12278:
12273:
12270:
12267:
12263:
12259:
12256:
12253:
12247:
12243:
12238:
12234:
12233:
12210:
12205:
12201:
12197:
12192:
12188:
12176:
12175:
12160:
12157:
12155:
12152:
12148:
12145:
12142:
12138:
12134:
12132:
12127:
12124:
12121:
12117:
12113:
12111:
12108:
12104:
12101:
12098:
12095:
12092:
12088:
12082:
12077:
12074:
12071:
12067:
12063:
12061:
12056:
12053:
12050:
12047:
12044:
12040:
12034:
12031:
12027:
12021:
12016:
12013:
12010:
12006:
12002:
12000:
11997:
11993:
11989:
11985:
11978:
11977:
11973:
11970:
11965:
11961:
11957:
11954:
11951:
11948:
11945:
11939:
11936:
11933:
11929:
11924:
11920:
11919:
11896:
11891:
11888:
11885:
11881:
11877:
11872:
11869:
11866:
11862:
11839:
11836:
11833:
11829:
11808:
11805:
11802:
11799:
11796:
11793:
11790:
11787:
11784:
11764:
11759:
11755:
11751:
11746:
11742:
11721:
11718:
11715:
11712:
11709:
11706:
11703:
11681:
11678:
11675:
11671:
11667:
11664:
11661:
11656:
11652:
11629:
11625:
11602:
11598:
11575:
11571:
11558:
11555:
11554:
11553:
11538:
11535:
11533:
11530:
11526:
11522:
11518:
11516:
11511:
11507:
11503:
11501:
11498:
11494:
11491:
11487:
11481:
11476:
11473:
11470:
11466:
11462:
11460:
11455:
11452:
11448:
11442:
11439:
11436:
11432:
11426:
11421:
11418:
11415:
11411:
11407:
11405:
11402:
11398:
11394:
11390:
11383:
11382:
11378:
11375:
11370:
11367:
11364:
11360:
11356:
11351:
11347:
11343:
11338:
11334:
11330:
11327:
11324:
11318:
11314:
11309:
11305:
11304:
11281:
11278:
11275:
11255:
11250:
11247:
11244:
11240:
11236:
11231:
11227:
11223:
11218:
11214:
11202:
11201:
11186:
11183:
11181:
11178:
11174:
11170:
11166:
11164:
11159:
11155:
11151:
11149:
11146:
11142:
11139:
11136:
11132:
11126:
11121:
11118:
11115:
11111:
11107:
11105:
11100:
11097:
11094:
11090:
11084:
11081:
11078:
11075:
11072:
11068:
11062:
11057:
11054:
11051:
11047:
11043:
11041:
11038:
11034:
11031:
11028:
11024:
11020:
11013:
11012:
11008:
11003:
11000:
10997:
10993:
10988:
10984:
10979:
10976:
10973:
10970:
10967:
10963:
10959:
10954:
10951:
10948:
10944:
10940:
10935:
10932:
10929:
10925:
10921:
10918:
10915:
10909:
10905:
10900:
10896:
10895:
10872:
10869:
10866:
10863:
10860:
10857:
10854:
10851:
10848:
10825:
10820:
10817:
10814:
10811:
10808:
10804:
10800:
10795:
10792:
10789:
10785:
10781:
10776:
10773:
10770:
10766:
10743:
10740:
10737:
10734:
10731:
10727:
10704:
10701:
10698:
10694:
10671:
10668:
10665:
10662:
10659:
10655:
10634:
10629:
10625:
10621:
10618:
10598:
10593:
10590:
10587:
10584:
10581:
10577:
10572:
10568:
10563:
10559:
10555:
10552:
10549:
10546:
10535:
10534:
10519:
10516:
10514:
10511:
10507:
10504:
10501:
10497:
10493:
10491:
10486:
10483:
10480:
10476:
10472:
10470:
10467:
10463:
10460:
10457:
10454:
10451:
10447:
10441:
10436:
10433:
10430:
10426:
10422:
10420:
10415:
10412:
10409:
10406:
10403:
10399:
10393:
10390:
10386:
10380:
10375:
10372:
10369:
10365:
10361:
10359:
10356:
10352:
10348:
10344:
10337:
10336:
10332:
10327:
10324:
10321:
10318:
10315:
10311:
10306:
10302:
10297:
10293:
10289:
10286:
10283:
10280:
10277:
10271:
10268:
10265:
10261:
10256:
10252:
10251:
10226:
10223:
10220:
10216:
10195:
10190:
10187:
10184:
10180:
10176:
10173:
10170:
10165:
10161:
10157:
10154:
10149:
10146:
10143:
10140:
10137:
10133:
10112:
10109:
10106:
10103:
10100:
10074:
10069:
10066:
10062:
10039:
10035:
10031:
10028:
10025:
10020:
10016:
9995:
9992:
9989:
9986:
9983:
9963:
9960:
9957:
9946:
9945:
9934:
9931:
9928:
9923:
9919:
9915:
9912:
9909:
9906:
9903:
9877:
9866:
9865:
9854:
9851:
9846:
9843:
9840:
9837:
9834:
9830:
9826:
9821:
9818:
9815:
9812:
9809:
9805:
9799:
9796:
9792:
9786:
9781:
9778:
9775:
9771:
9767:
9762:
9758:
9732:
9728:
9707:
9704:
9701:
9698:
9695:
9692:
9689:
9678:
9677:
9666:
9661:
9657:
9653:
9650:
9645:
9642:
9639:
9635:
9631:
9626:
9623:
9619:
9613:
9608:
9605:
9602:
9598:
9574:
9571:
9568:
9565:
9562:
9559:
9556:
9553:
9550:
9530:
9527:
9524:
9521:
9518:
9515:
9512:
9492:
9489:
9486:
9481:
9478:
9475:
9471:
9467:
9462:
9459:
9455:
9424:
9420:
9399:
9396:
9393:
9390:
9387:
9384:
9381:
9378:
9375:
9355:
9350:
9347:
9344:
9340:
9336:
9331:
9327:
9323:
9318:
9314:
9293:
9273:
9268:
9264:
9260:
9257:
9254:
9249:
9245:
9241:
9238:
9233:
9230:
9227:
9223:
9202:
9197:
9194:
9191:
9187:
9183:
9178:
9174:
9170:
9165:
9161:
9151:is a function
9140:
9135:
9132:
9128:
9124:
9121:
9118:
9113:
9110:
9106:
9102:
9099:
9094:
9090:
9069:
9049:
9044:
9040:
9036:
9031:
9027:
9023:
9018:
9014:
8991:
8987:
8966:
8963:
8960:
8939:
8934:
8931:
8927:
8923:
8920:
8917:
8912:
8908:
8904:
8901:
8896:
8892:
8871:
8868:
8865:
8843:
8839:
8835:
8832:
8829:
8824:
8820:
8799:
8796:
8793:
8790:
8787:
8784:
8781:
8761:
8756:
8753:
8749:
8745:
8742:
8739:
8734:
8731:
8727:
8723:
8720:
8715:
8711:
8685:
8681:
8677:
8672:
8669:
8665:
8659:
8654:
8651:
8648:
8644:
8621:
8618:
8614:
8593:
8588:
8585:
8581:
8577:
8574:
8571:
8566:
8562:
8558:
8555:
8550:
8546:
8525:
8503:
8499:
8478:
8458:
8455:
8452:
8449:
8446:
8443:
8440:
8437:
8434:
8414:
8392:
8388:
8367:
8364:
8361:
8340:Main article:
8337:
8334:
8322:
8319:
8287:
8284:
8282:
8279:
8266:
8261:
8256:
8253:
8233:
8230:
8227:
8224:
8204:
8201:
8198:
8193:
8189:
8164:
8159:
8155:
8150:
8146:
8142:
8139:
8136:
8133:
8128:
8123:
8120:
8117:
8113:
8107:
8104:
8099:
8096:
8091:
8087:
8083:
8080:
8077:
8074:
8070:
8063:
8058:
8055:
8052:
8048:
8041:
8038:
8035:
8031:
8026:
8023:
8020:
8017:
8012:
8005:
8002:
7989:
7976:
7973:
7970:
7967:
7947:
7944:
7940:
7936:
7933:
7930:
7927:
7922:
7919:
7915:
7911:
7906:
7902:
7898:
7894:
7870:
7863:
7858:
7855:
7852:
7846:
7843:
7832:
7828:
7824:
7820:
7816:
7813:
7810:
7807:
7802:
7795:
7792:
7785:
7779:
7774:
7771:
7768:
7762:
7759:
7748:
7744:
7740:
7736:
7732:
7729:
7726:
7723:
7718:
7711:
7708:
7700:
7690:
7677:
7674:
7671:
7668:
7648:
7645:
7642:
7639:
7636:
7633:
7613:
7610:
7607:
7602:
7598:
7592:
7589:
7567:
7564:
7561:
7558:
7538:
7535:
7532:
7527:
7520:
7517:
7493:
7469:
7466:
7463:
7458:
7451:
7448:
7421:
7418:
7415:
7412:
7407:
7403:
7399:
7396:
7393:
7388:
7366:
7363:
7360:
7355:
7351:
7330:
7308:
7304:
7275:
7270:
7244:
7240:
7232:
7227:
7222:
7219:
7216:
7213:
7210:
7207:
7202:
7195:
7192:
7185:
7180:
7169:
7152:
7149:
7146:
7143:
7140:
7137:
7134:
7131:
7128:
7125:
7122:
7119:
7116:
7113:
7110:
7105:
7101:
7080:
7077:
7074:
7069:
7065:
7059:
7056:
7034:
7031:
7028:
7025:
7005:
7002:
6999:
6994:
6987:
6984:
6960:
6957:
6954:
6932:
6928:
6924:
6921:
6918:
6913:
6909:
6896:
6893:
6892:
6891:
6890:
6889:
6877:
6874:
6871:
6851:
6848:
6845:
6840:
6833:
6830:
6823:
6818:
6814:
6810:
6806:
6783:
6779:
6775:
6770:
6763:
6760:
6742:
6741:
6740:
6739:
6727:
6724:
6719:
6715:
6692:
6688:
6684:
6679:
6675:
6652:
6648:
6644:
6641:
6630:
6619:
6616:
6613:
6593:
6571:
6567:
6544:
6540:
6536:
6531:
6527:
6515:
6503:
6500:
6495:
6488:
6485:
6459:
6452:
6449:
6425:
6414:
6403:
6400:
6397:
6377:
6374:
6371:
6351:
6348:
6345:
6342:
6322:
6319:
6316:
6311:
6304:
6301:
6286:
6275:
6255:
6252:
6249:
6246:
6235:
6224:
6204:
6179:
6174:
6169:
6166:
6140:
6135:
6111:
6107:
6084:
6081:
6078:
6073:
6066:
6063:
6052:
6048:
6044:
6041:
6037:
6027:
6014:
5992:
5989:
5985:
5982:
5978:
5975:
5972:
5967:
5964:
5960:
5957:
5952:
5948:
5943:
5940:
5937:
5933:
5929:
5926:
5923:
5920:
5917:
5914:
5910:
5900:
5899:
5898:
5897:
5896:
5883:
5863:
5841:
5838:
5835:
5832:
5829:
5825:
5804:
5801:
5798:
5778:
5775:
5772:
5767:
5760:
5757:
5750:
5745:
5741:
5737:
5733:
5710:
5706:
5702:
5697:
5690:
5687:
5669:
5668:
5667:
5666:
5654:
5651:
5646:
5639:
5636:
5610:
5603:
5600:
5576:
5565:
5554:
5551:
5548:
5528:
5525:
5522:
5502:
5499:
5496:
5493:
5473:
5470:
5467:
5462:
5455:
5452:
5437:
5426:
5406:
5403:
5400:
5397:
5386:
5375:
5355:
5330:
5325:
5320:
5317:
5306:
5294:
5291:
5288:
5266:
5262:
5239:
5232:
5229:
5205:
5185:
5182:
5179:
5159:
5156:
5153:
5150:
5130:
5127:
5124:
5119:
5112:
5109:
5094:
5093:
5092:
5081:
5061:
5058:
5055:
5052:
5032:
5029:
5026:
5021:
5014:
5011:
4987:
4967:
4964:
4961:
4958:
4947:
4936:
4931:
4928:
4923:
4920:
4900:
4895:
4891:
4887:
4882:
4875:
4872:
4846:
4843:
4832:converging to
4821:
4818:
4815:
4810:
4806:
4802:
4782:
4779:
4774:
4771:
4745:
4741:
4738:
4735:
4730:
4723:
4720:
4695:
4691:
4688:
4685:
4682:
4656:
4649:
4646:
4620:
4613:
4610:
4584:
4580:
4557:
4553:
4532:
4520:
4517:
4504:
4501:
4498:
4478:
4475:
4472:
4469:
4445:
4442:
4439:
4434:
4427:
4424:
4400:
4397:
4394:
4391:
4388:
4385:
4382:
4379:
4376:
4371:
4364:
4361:
4354:
4351:
4331:
4328:
4325:
4305:
4302:
4299:
4296:
4293:
4290:
4287:
4284:
4281:
4261:
4256:
4252:
4248:
4245:
4242:
4239:
4234:
4229:
4226:
4223:
4219:
4213:
4210:
4182:
4179:
4174:
4170:
4166:
4163:
4160:
4157:
4152:
4147:
4144:
4141:
4137:
4131:
4128:
4123:
4120:
4117:
4114:
4111:
4108:
4105:
4102:
4099:
4094:
4087:
4084:
4077:
4072:
4069:
4066:
4062:
4052:
4036:
4016:
3996:
3976:
3954:
3950:
3946:
3943:
3940:
3935:
3931:
3907:
3904:
3901:
3898:
3895:
3892:
3872:
3869:
3866:
3846:
3843:
3840:
3816:
3813:
3810:
3807:
3784:
3781:
3778:
3775:
3772:
3769:
3748:
3744:
3741:
3738:
3735:
3732:
3729:
3707:
3702:
3697:
3694:
3674:
3654:
3632:
3627:
3605:
3583:
3580:
3577:
3574:
3571:
3568:
3565:
3562:
3559:
3556:
3553:
3550:
3547:
3544:
3541:
3538:
3535:
3532:
3529:
3526:
3523:
3518:
3515:
3512:
3508:
3498:
3493:
3490:
3475:
3472:
3467:
3462:
3458:
3437:
3434:
3431:
3428:
3425:
3422:
3419:
3397:
3393:
3370:
3366:
3362:
3359:
3356:
3351:
3347:
3343:
3338:
3334:
3303:
3300:
3298:
3295:
3293:
3290:
3287:
3284:
3281:
3279:
3276:
3274:
3271:
3268:
3261:
3259:
3255:
3250:
3246:
3242:
3239:
3236:
3233:
3228:
3223:
3220:
3217:
3213:
3207:
3204:
3199:
3196:
3191:
3187:
3183:
3177:
3172:
3167:
3164:
3160:
3156:
3154:
3151:
3148:
3145:
3140:
3133:
3130:
3123:
3122:
3098:
3093:
3089:
3085:
3082:
3079:
3076:
3071:
3066:
3063:
3060:
3056:
3050:
3047:
3042:
3039:
3036:
3033:
3028:
3021:
3018:
2992:
2989:
2986:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2929:
2909:
2887:
2883:
2879:
2876:
2873:
2868:
2864:
2860:
2855:
2851:
2837:
2834:
2821:
2801:
2775:
2771:
2767:
2764:
2743:
2723:
2698:
2694:
2676:
2673:
2660:
2640:
2620:
2600:
2580:
2558:
2554:
2527:
2524:
2522:
2519:
2515:
2511:
2507:
2505:
2502:
2500:
2497:
2495:
2492:
2488:
2484:
2480:
2478:
2475:
2471:
2467:
2463:
2461:
2458:
2456:
2453:
2451:
2448:
2446:
2443:
2441:
2436:
2432:
2428:
2426:
2423:
2419:
2415:
2409:
2405:
2401:
2399:
2396:
2388:
2383:
2379:
2375:
2371:
2368:
2365:
2360:
2357:
2350:
2347:
2343:
2338:
2334:
2330:
2328:
2325:
2317:
2313:
2307:
2303:
2299:
2297:
2294:
2290:
2285:
2281:
2277:
2273:
2268:
2264:
2260:
2258:
2255:
2245:
2241:
2235:
2231:
2227:
2225:
2222:
2220:
2215:
2211:
2207:
2203:
2200:
2197:
2195:
2192:
2182:
2179:
2176:
2174:
2171:
2169:
2166:
2163:
2156:
2155:
2148:
2144:
2138:
2133:
2129:
2123:
2119:
2115:
2113:
2110:
2108:
2105:
2103:
2100:
2096:
2092:
2086:
2081:
2077:
2071:
2067:
2063:
2061:
2058:
2054:
2050:
2044:
2039:
2035:
2029:
2025:
2021:
2019:
2016:
2014:
2009:
2005:
2001:
1999:
1996:
1994:
1989:
1985:
1981:
1974:
1973:
1951:
1948:
1945:
1942:
1939:
1936:
1933:
1911:
1907:
1893:
1890:
1874:
1871:
1867:
1844:
1841:
1837:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1772:
1768:
1747:
1727:
1701:
1698:
1696:
1693:
1689:
1685:
1681:
1679:
1676:
1674:
1671:
1669:
1666:
1664:
1661:
1659:
1654:
1650:
1646:
1644:
1641:
1637:
1633:
1627:
1623:
1619:
1617:
1614:
1612:
1607:
1603:
1599:
1595:
1592:
1589:
1587:
1584:
1580:
1577:
1574:
1572:
1569:
1567:
1564:
1561:
1554:
1553:
1546:
1542:
1536:
1531:
1527:
1523:
1521:
1518:
1516:
1511:
1507:
1503:
1501:
1498:
1496:
1491:
1487:
1483:
1476:
1475:
1451:
1448:
1444:
1421:
1418:
1414:
1401:linear program
1396:
1393:
1389:
1388:
1385:
1366:
1346:
1319:
1314:
1310:
1306:
1303:
1300:
1297:
1292:
1288:
1282:
1277:
1274:
1271:
1267:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1214:
1210:
1206:
1203:
1200:
1195:
1191:
1168:
1164:
1160:
1157:
1154:
1149:
1145:
1120:
1108:
1107:Discretization
1105:
1103:
1100:
1087:
1067:
1046:
1020:
1008:
1005:
984:
979:
975:
954:
951:
948:
945:
925:
922:
901:
896:
892:
868:
848:
828:
808:
805:
802:
799:
796:
793:
790:
787:
784:
781:
759:
754:
749:
746:
724:
719:
714:
711:
684:
681:
678:
675:
673:
669:
666:
663:
660:
657:
654:
651:
648:
645:
642:
639:
636:
633:
630:
627:
624:
621:
614:
613:
609:
604:
600:
596:
593:
590:
587:
581:
576:
571:
568:
564:
560:
559:
539:
536:
533:
530:
527:
524:
497:
494:
491:
488:
486:
482:
479:
476:
473:
470:
463:
462:
458:
455:
452:
449:
446:
443:
440:
437:
432:
428:
424:
421:
416:
412:
408:
405:
402:
399:
396:
393:
387:
382:
377:
374:
370:
366:
365:
343:
340:
337:
334:
331:
328:
325:
322:
319:
316:
313:
310:
307:
304:
301:
298:
295:
292:
287:
282:
279:
276:
273:
270:
267:
264:
259:
255:
234:
231:
228:
225:
222:
219:
199:
196:
193:
190:
187:
184:
181:
178:
175:
170:
166:
162:
159:
156:
153:
150:
147:
144:
141:
138:
135:
132:
127:
124:
121:
117:
103:
100:
99:
98:
93:
88:
83:
77:
74:
66:
63:
59:transportation
15:
9:
6:
4:
3:
2:
13255:
13244:
13241:
13239:
13236:
13235:
13233:
13218:
13215:
13213:
13210:
13208:
13205:
13203:
13200:
13198:
13195:
13193:
13190:
13188:
13185:
13183:
13180:
13178:
13175:
13173:
13170:
13168:
13165:
13163:
13160:
13158:
13155:
13154:
13151:
13147:
13140:
13135:
13133:
13128:
13126:
13121:
13120:
13117:
13111:
13108:
13107:
13097:
13091:
13087:
13086:
13080:
13077:
13073:
13069:
13064:|agency=
13056:
13042:on 2020-03-24
13038:
13034:
13030:
13026:
13020:
13013:
13012:
13007:
13003:
12998:
12995:
12991:
12988:
12985:
12982:
12979:
12978:
12973:
12969:
12965:
12961:
12959:0-471-95158-7
12955:
12951:
12950:
12944:
12941:
12940:
12935:
12934:
12923:
12917:
12911:
12910:0-521-65539-0
12907:
12903:
12897:
12891:
12890:0-691-08506-4
12887:
12883:
12877:
12869:
12863:
12859:
12855:
12851:
12845:
12837:
12831:
12820:
12819:
12811:
12804:
12798:
12791:
12790:
12783:
12775:
12771:
12767:
12761:
12757:
12753:
12749:
12748:
12740:
12732:
12727:|agency=
12719:
12705:on 2020-03-24
12701:
12697:
12693:
12689:
12683:
12676:
12675:
12670:
12666:
12659:
12655:
12646:
12643:
12641:
12638:
12636:
12633:
12631:
12628:
12626:
12623:
12621:
12618:
12616:
12613:
12611:
12608:
12607:
12601:
12595:
12591:
12587:
12584:
12581:
12577:
12576:Value at risk
12573:
12569:
12566:
12563:
12560:
12559:
12558:
12556:
12541:
12527:
12524:
12521:
12518:
12515:
12512:
12509:
12506:
12503:
12476:
12471:
12464:
12460:
12449:
12445:
12439:
12432:
12429:
12426:
12422:
12416:
12411:
12408:
12405:
12401:
12390:
12387:
12384:
12380:
12374:
12371:
12368:
12365:
12362:
12358:
12352:
12347:
12344:
12341:
12337:
12331:
12324:
12321:
12318:
12314:
12290:
12287:
12284:
12280:
12271:
12268:
12265:
12261:
12254:
12245:
12241:
12224:
12223:
12222:
12203:
12199:
12190:
12186:
12158:
12153:
12146:
12143:
12140:
12136:
12125:
12122:
12119:
12115:
12109:
12102:
12099:
12096:
12093:
12090:
12086:
12080:
12075:
12072:
12069:
12065:
12054:
12051:
12048:
12045:
12042:
12038:
12032:
12029:
12025:
12019:
12014:
12011:
12008:
12004:
11998:
11991:
11987:
11963:
11959:
11952:
11946:
11937:
11934:
11931:
11927:
11910:
11909:
11908:
11889:
11886:
11883:
11879:
11870:
11867:
11864:
11860:
11834:
11827:
11806:
11803:
11800:
11797:
11794:
11791:
11788:
11785:
11782:
11757:
11753:
11744:
11740:
11719:
11716:
11713:
11710:
11707:
11704:
11701:
11679:
11676:
11673:
11669:
11665:
11662:
11659:
11654:
11650:
11627:
11623:
11600:
11596:
11573:
11569:
11536:
11531:
11524:
11520:
11509:
11505:
11499:
11492:
11489:
11485:
11479:
11474:
11471:
11468:
11464:
11453:
11450:
11446:
11440:
11437:
11434:
11430:
11424:
11419:
11416:
11413:
11409:
11403:
11396:
11392:
11365:
11358:
11354:
11349:
11345:
11336:
11332:
11325:
11316:
11312:
11295:
11294:
11293:
11279:
11276:
11273:
11245:
11238:
11234:
11229:
11225:
11216:
11212:
11184:
11179:
11172:
11168:
11157:
11153:
11147:
11140:
11137:
11134:
11130:
11124:
11119:
11116:
11113:
11109:
11098:
11095:
11092:
11088:
11082:
11079:
11076:
11073:
11070:
11066:
11060:
11055:
11052:
11049:
11045:
11039:
11032:
11029:
11026:
11022:
10998:
10991:
10974:
10971:
10968:
10961:
10957:
10952:
10949:
10946:
10942:
10933:
10930:
10927:
10923:
10916:
10907:
10903:
10886:
10885:
10884:
10870:
10867:
10864:
10861:
10858:
10855:
10852:
10849:
10846:
10837:
10815:
10812:
10809:
10802:
10798:
10793:
10790:
10787:
10783:
10774:
10771:
10768:
10764:
10738:
10735:
10732:
10725:
10702:
10699:
10696:
10692:
10666:
10663:
10660:
10653:
10627:
10623:
10616:
10588:
10585:
10582:
10575:
10561:
10557:
10550:
10544:
10517:
10512:
10505:
10502:
10499:
10495:
10484:
10481:
10478:
10474:
10468:
10461:
10458:
10455:
10452:
10449:
10445:
10439:
10434:
10431:
10428:
10424:
10413:
10410:
10407:
10404:
10401:
10397:
10391:
10388:
10384:
10378:
10373:
10370:
10367:
10363:
10357:
10350:
10346:
10322:
10319:
10316:
10309:
10295:
10291:
10284:
10278:
10269:
10266:
10263:
10259:
10242:
10241:
10240:
10224:
10221:
10218:
10214:
10188:
10185:
10182:
10178:
10174:
10171:
10168:
10163:
10159:
10152:
10144:
10141:
10138:
10131:
10110:
10107:
10104:
10101:
10098:
10090:
10085:
10072:
10067:
10064:
10060:
10037:
10033:
10029:
10026:
10023:
10018:
10014:
9993:
9990:
9987:
9984:
9981:
9961:
9958:
9955:
9932:
9921:
9917:
9910:
9904:
9894:
9893:
9892:
9889:
9875:
9852:
9841:
9838:
9835:
9828:
9819:
9816:
9813:
9810:
9807:
9803:
9797:
9794:
9790:
9784:
9779:
9776:
9773:
9769:
9765:
9760:
9756:
9748:
9747:
9746:
9730:
9726:
9705:
9702:
9699:
9696:
9693:
9690:
9687:
9664:
9659:
9655:
9651:
9640:
9633:
9624:
9621:
9617:
9611:
9606:
9603:
9600:
9596:
9588:
9587:
9586:
9572:
9569:
9566:
9563:
9560:
9557:
9554:
9551:
9548:
9528:
9525:
9522:
9519:
9516:
9513:
9510:
9490:
9487:
9476:
9469:
9460:
9457:
9453:
9444:
9440:
9422:
9418:
9397:
9394:
9391:
9388:
9385:
9382:
9379:
9376:
9373:
9345:
9338:
9329:
9325:
9321:
9316:
9312:
9291:
9266:
9262:
9258:
9255:
9252:
9247:
9243:
9236:
9228:
9221:
9192:
9185:
9176:
9172:
9168:
9163:
9159:
9133:
9130:
9126:
9122:
9119:
9116:
9111:
9108:
9104:
9097:
9092:
9088:
9080:the decision
9067:
9042:
9038:
9029:
9025:
9021:
9016:
9012:
8989:
8985:
8964:
8961:
8958:
8932:
8929:
8925:
8921:
8918:
8915:
8910:
8906:
8899:
8894:
8890:
8869:
8866:
8863:
8841:
8837:
8833:
8830:
8827:
8822:
8818:
8797:
8794:
8791:
8788:
8785:
8782:
8779:
8754:
8751:
8747:
8743:
8740:
8737:
8732:
8729:
8725:
8718:
8713:
8709:
8699:
8698:should hold.
8683:
8679:
8675:
8670:
8667:
8663:
8657:
8652:
8649:
8646:
8642:
8619:
8616:
8612:
8586:
8583:
8579:
8575:
8572:
8569:
8564:
8560:
8553:
8548:
8544:
8523:
8501:
8497:
8476:
8456:
8453:
8450:
8447:
8444:
8441:
8438:
8435:
8432:
8412:
8405:to invest in
8390:
8386:
8365:
8362:
8359:
8349:
8343:
8333:
8331:
8327:
8318:
8316:
8312:
8308:
8304:
8300:
8296:
8292:
8278:
8259:
8251:
8228:
8222:
8199:
8191:
8187:
8162:
8157:
8148:
8144:
8140:
8137:
8131:
8126:
8121:
8118:
8115:
8111:
8105:
8102:
8097:
8089:
8085:
8081:
8078:
8072:
8068:
8061:
8056:
8053:
8050:
8046:
8039:
8036:
8033:
8029:
8024:
8018:
8010:
8000:
7988:
7971:
7942:
7938:
7934:
7931:
7928:
7920:
7917:
7909:
7904:
7900:
7896:
7892:
7868:
7861:
7853:
7841:
7830:
7826:
7822:
7818:
7814:
7808:
7800:
7790:
7783:
7777:
7769:
7757:
7746:
7742:
7738:
7734:
7730:
7724:
7716:
7706:
7698:
7689:
7672:
7666:
7643:
7640:
7637:
7631:
7608:
7600:
7596:
7590:
7587:
7578:and variance
7562:
7556:
7533:
7525:
7515:
7491:
7483:
7464:
7456:
7446:
7433:
7413:
7405:
7401:
7397:
7394:
7377:, written as
7361:
7353:
7349:
7341:and variance
7328:
7306:
7302:
7293:
7268:
7242:
7238:
7225:
7214:
7208:
7205:
7200:
7190:
7178:
7168:
7167:we have that
7166:
7144:
7141:
7138:
7132:
7126:
7123:
7120:
7117:
7111:
7103:
7099:
7075:
7067:
7063:
7057:
7054:
7029:
7023:
7000:
6992:
6982:
6958:
6955:
6952:
6930:
6926:
6922:
6919:
6916:
6911:
6907:
6869:
6849:
6838:
6828:
6821:
6816:
6812:
6781:
6777:
6768:
6758:
6746:
6745:
6744:
6743:
6725:
6717:
6713:
6690:
6686:
6682:
6677:
6673:
6650:
6646:
6642:
6639:
6631:
6617:
6614:
6611:
6591:
6569:
6565:
6542:
6538:
6534:
6529:
6525:
6516:
6501:
6498:
6493:
6483:
6457:
6447:
6423:
6415:
6401:
6398:
6395:
6369:
6346:
6340:
6333:converges to
6317:
6309:
6299:
6287:
6273:
6250:
6244:
6237:the function
6236:
6222:
6202:
6194:
6193:
6177:
6167:
6164:
6156:
6155:
6154:
6138:
6109:
6105:
6079:
6071:
6061:
6050:
6046:
6042:
6039:
6026:
6012:
5983:
5980:
5976:
5973:
5965:
5962:
5958:
5955:
5941:
5938:
5935:
5927:
5921:
5918:
5915:
5894:
5881:
5861:
5836:
5833:
5830:
5796:
5776:
5765:
5755:
5748:
5743:
5739:
5708:
5704:
5695:
5685:
5673:
5672:
5671:
5670:
5652:
5649:
5644:
5634:
5608:
5598:
5574:
5566:
5552:
5549:
5546:
5520:
5497:
5491:
5484:converges to
5468:
5460:
5450:
5438:
5424:
5401:
5395:
5388:the function
5387:
5373:
5353:
5345:
5344:
5328:
5318:
5315:
5307:
5286:
5264:
5260:
5252:converges to
5237:
5227:
5203:
5177:
5154:
5148:
5125:
5117:
5107:
5095:
5079:
5056:
5050:
5043:converges to
5027:
5019:
5009:
4985:
4962:
4956:
4949:the function
4948:
4926:
4918:
4911:converges to
4893:
4889:
4880:
4870:
4841:
4819:
4816:
4808:
4804:
4780:
4777:
4769:
4759:
4758:
4736:
4733:
4728:
4718:
4686:
4683:
4680:
4672:
4671:
4670:
4654:
4644:
4618:
4608:
4582:
4578:
4555:
4551:
4530:
4516:
4496:
4473:
4467:
4460:estimator of
4459:
4440:
4432:
4422:
4395:
4389:
4386:
4377:
4369:
4359:
4349:
4323:
4297:
4294:
4291:
4285:
4279:
4254:
4250:
4246:
4243:
4237:
4232:
4227:
4224:
4221:
4217:
4211:
4208:
4198:
4172:
4168:
4164:
4161:
4155:
4150:
4145:
4142:
4139:
4135:
4129:
4126:
4121:
4115:
4109:
4106:
4100:
4092:
4082:
4070:
4067:
4064:
4051:
4050:
4034:
4014:
3994:
3974:
3952:
3948:
3944:
3941:
3938:
3933:
3929:
3919:
3902:
3899:
3896:
3890:
3870:
3867:
3864:
3844:
3841:
3838:
3830:
3829:finite valued
3811:
3805:
3796:
3779:
3776:
3773:
3767:
3736:
3733:
3730:
3727:
3705:
3695:
3672:
3652:
3630:
3603:
3572:
3569:
3566:
3560:
3554:
3551:
3545:
3539:
3536:
3530:
3524:
3516:
3513:
3510:
3497:
3489:
3473:
3470:
3465:
3460:
3456:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3395:
3391:
3368:
3364:
3360:
3357:
3354:
3349:
3345:
3341:
3336:
3332:
3323:
3318:
3301:
3296:
3291:
3282:
3277:
3272:
3269:
3248:
3244:
3240:
3237:
3231:
3226:
3221:
3218:
3215:
3211:
3205:
3202:
3197:
3194:
3189:
3185:
3175:
3165:
3162:
3152:
3146:
3138:
3128:
3112:
3109:
3091:
3087:
3083:
3080:
3074:
3069:
3064:
3061:
3058:
3054:
3048:
3045:
3040:
3034:
3026:
3016:
3004:
2984:
2981:
2978:
2972:
2966:
2963:
2957:
2951:
2943:
2927:
2907:
2885:
2881:
2877:
2874:
2871:
2866:
2862:
2858:
2853:
2849:
2833:
2819:
2799:
2791:
2773:
2769:
2765:
2762:
2741:
2721:
2712:
2696:
2692:
2683:
2672:
2658:
2638:
2618:
2598:
2578:
2556:
2552:
2542:
2525:
2520:
2513:
2509:
2503:
2498:
2493:
2486:
2482:
2476:
2469:
2465:
2459:
2454:
2449:
2444:
2434:
2430:
2424:
2417:
2413:
2407:
2403:
2397:
2386:
2381:
2377:
2366:
2358:
2348:
2336:
2332:
2326:
2315:
2311:
2305:
2301:
2295:
2288:
2283:
2279:
2266:
2262:
2256:
2243:
2239:
2233:
2229:
2223:
2218:
2213:
2209:
2198:
2193:
2180:
2177:
2172:
2167:
2164:
2146:
2142:
2131:
2127:
2121:
2117:
2111:
2106:
2101:
2094:
2090:
2084:
2079:
2075:
2069:
2065:
2059:
2052:
2048:
2037:
2033:
2027:
2023:
2017:
2012:
2003:
1997:
1992:
1983:
1963:
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1909:
1905:
1889:
1872:
1869:
1865:
1842:
1839:
1835:
1825:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1770:
1766:
1745:
1725:
1716:
1699:
1694:
1687:
1683:
1677:
1672:
1667:
1662:
1652:
1648:
1642:
1635:
1631:
1625:
1621:
1615:
1610:
1605:
1601:
1590:
1585:
1578:
1575:
1570:
1565:
1562:
1544:
1540:
1534:
1529:
1525:
1519:
1514:
1509:
1505:
1499:
1494:
1489:
1485:
1465:
1449:
1446:
1442:
1419:
1416:
1412:
1402:
1399:A stochastic
1392:
1386:
1383:
1379:
1375:
1371:
1367:
1364:
1360:
1359:
1358:
1344:
1335:
1333:
1312:
1308:
1304:
1301:
1295:
1290:
1286:
1280:
1275:
1272:
1269:
1261:
1252:
1249:
1246:
1240:
1234:
1212:
1208:
1204:
1201:
1198:
1193:
1189:
1166:
1162:
1158:
1155:
1152:
1147:
1143:
1134:
1118:
1099:
1085:
1065:
1044:
1036:
1035:
1018:
1004:
1001:
996:
982:
977:
973:
952:
949:
946:
943:
923:
920:
899:
894:
890:
880:
866:
846:
826:
803:
800:
797:
794:
791:
788:
785:
779:
757:
747:
744:
722:
712:
709:
700:
682:
679:
676:
664:
658:
655:
652:
646:
640:
637:
634:
628:
622:
607:
602:
594:
588:
579:
569:
566:
534:
531:
528:
522:
513:
495:
492:
489:
480:
477:
474:
471:
450:
447:
444:
438:
430:
426:
422:
419:
414:
410:
406:
400:
394:
385:
375:
372:
354:
341:
332:
326:
323:
320:
314:
308:
305:
302:
296:
290:
277:
274:
271:
265:
257:
229:
226:
223:
217:
188:
185:
182:
176:
168:
164:
160:
154:
148:
145:
139:
133:
125:
122:
119:
97:
94:
92:
89:
87:
84:
81:
78:
75:
72:
71:
70:
62:
60:
56:
52:
48:
44:
40:
37:
33:
29:
22:
13181:
13084:
13075:
13044:. Retrieved
13037:the original
13010:
12993:
12975:
12948:
12937:
12916:
12901:
12896:
12881:
12876:
12853:
12844:
12830:
12817:
12810:
12797:
12787:
12782:
12746:
12739:
12707:. Retrieved
12700:the original
12673:
12658:
12599:
12552:
12495:
12177:
11560:
11203:
10838:
10536:
10086:
9947:
9890:
9867:
9745:is given by
9679:
9442:
9438:
8700:
8536:assets. Let
8351:
8324:
8307:life-history
8289:
8178:
7883:
7481:
7434:
7292:distribution
7291:
7258:
6898:
6096:
6004:
5895:, defined as
5853:
5852:denotes the
5815:. Note that
4522:
4457:
4194:
4048:
3920:
3828:
3798:Assume that
3797:
3595:
3495:
3321:
3319:
3113:
3110:
3005:
2839:
2789:
2713:
2681:
2678:
2543:
1964:
1895:
1826:
1718:The vectors
1717:
1466:
1398:
1390:
1336:
1132:
1110:
1033:
1032:
1010:
999:
997:
881:
701:
514:
355:
105:
68:
46:
39:optimization
31:
25:
11852:. That is,
9718:the wealth
2682:first stage
43:uncertainty
13232:Categories
13046:2010-09-22
12709:2010-09-22
12651:References
12308:subject to
11981:subject to
11386:subject to
11016:subject to
10340:subject to
8516:among the
8346:See also:
8315:parasitoid
6705:such that
6192:such that
5343:such that
3883:the value
3264:subject to
2159:subject to
1557:subject to
617:subject to
466:subject to
13066:ignored (
13055:cite book
12774:1431-8598
12729:ignored (
12718:cite book
12596:problems)
12522:…
12513:−
12472:≥
12402:∑
12359:ξ
12338:∑
12154:≥
12144:−
12123:−
12100:−
12066:∑
12052:−
12026:ξ
12005:∑
11935:−
11887:−
11868:−
11828:ξ
11804:−
11795:…
11714:…
11677:−
11670:ξ
11663:…
11651:ξ
11624:ξ
11597:ξ
11570:ξ
11532:≥
11465:∑
11431:ξ
11410:∑
11359:ξ
11239:ξ
11180:≥
11110:∑
11067:ξ
11046:∑
10992:ξ
10962:ξ
10865:…
10856:−
10813:−
10803:ξ
10791:−
10772:−
10736:−
10726:ξ
10700:−
10664:−
10654:ξ
10586:−
10576:ξ
10513:≥
10503:−
10482:−
10459:−
10425:∑
10411:−
10385:ξ
10364:∑
10320:−
10310:ξ
10267:−
10222:−
10186:−
10179:ξ
10172:…
10160:ξ
10142:−
10132:ξ
10108:−
10034:ξ
10027:…
10015:ξ
9991:−
9839:−
9829:ξ
9817:−
9791:ξ
9770:∑
9700:…
9634:ξ
9597:∑
9570:−
9561:…
9523:…
9488:≥
9470:ξ
9395:−
9386:…
9339:ξ
9263:ξ
9256:…
9244:ξ
9222:ξ
9186:ξ
9120:…
9039:ξ
8986:ξ
8919:…
8838:ξ
8831:…
8819:ξ
8792:…
8748:ξ
8741:…
8726:ξ
8710:ξ
8643:∑
8573:…
8454:−
8445:…
8188:σ
8145:ξ
8112:∑
8098:−
8086:ξ
8047:∑
8037:−
8004:^
8001:σ
7972:⋅
7966:Φ
7935:α
7932:−
7918:−
7914:Φ
7897:α
7845:^
7842:σ
7823:α
7794:^
7761:^
7758:σ
7739:α
7731:−
7710:^
7644:α
7641:−
7597:σ
7519:^
7450:^
7402:σ
7350:σ
7206:−
7194:^
7145:ξ
7100:σ
7064:σ
6986:^
6956:∈
6927:ξ
6920:…
6908:ξ
6876:∞
6873:→
6847:→
6832:^
6817:∗
6782:∗
6778:ϑ
6774:→
6762:^
6759:ϑ
6723:→
6683:∈
6651:∗
6643:∈
6615:∈
6535:∈
6499:⊂
6487:^
6451:^
6399:∈
6376:∞
6373:→
6303:^
6168:⊂
6065:^
6043:∈
5988:‖
5977:−
5971:‖
5963:∈
5939:∈
5874:from set
5803:∞
5800:→
5774:→
5759:^
5744:∗
5709:∗
5705:ϑ
5701:→
5689:^
5686:ϑ
5650:⊂
5638:^
5602:^
5550:∈
5527:∞
5524:→
5454:^
5319:⊂
5293:∞
5290:→
5265:∗
5261:ϑ
5231:^
5228:ϑ
5184:∞
5181:→
5111:^
5057:⋅
5028:⋅
5013:^
4963:⋅
4930:¯
4874:^
4845:¯
4817:⊂
4778:∈
4773:¯
4740:→
4722:^
4690:→
4648:^
4612:^
4609:ϑ
4583:∗
4556:∗
4552:ϑ
4503:∞
4500:→
4426:^
4363:^
4330:∞
4327:→
4298:ξ
4251:ξ
4218:∑
4169:ξ
4136:∑
4086:^
4068:∈
4035:ξ
3995:ξ
3949:ξ
3942:…
3930:ξ
3903:ξ
3868:∈
3842:∈
3780:ξ
3743:→
3740:Ξ
3737:×
3696:⊂
3693:Ξ
3653:ξ
3573:ξ
3514:∈
3430:…
3392:ξ
3365:ξ
3358:…
3346:ξ
3333:ξ
3297:≥
3245:ξ
3212:∑
3166:∈
3132:^
3088:ξ
3055:∑
3020:^
2985:ξ
2928:ξ
2882:ξ
2875:…
2863:ξ
2850:ξ
2820:ξ
2734:contains
2722:ξ
2697:∗
2521:≥
2499:…
2367:⋮
2359:⋱
2349:⋮
2137:⊤
2107:⋯
2043:⊤
2008:⊤
1988:⊤
1944:…
1695:≥
1345:ξ
1309:ξ
1266:∑
1253:ξ
1202:…
1163:ξ
1156:…
1144:ξ
1133:scenarios
1119:ξ
1086:ξ
1066:ξ
1045:ξ
1019:ξ
950:≤
867:ξ
847:ξ
780:ξ
748:∈
713:∈
680:≥
665:ξ
647:ξ
629:ξ
595:ξ
570:∈
535:ξ
493:≥
451:ξ
431:ξ
376:∈
333:ξ
315:ξ
297:ξ
278:ξ
230:ξ
189:ξ
169:ξ
123:∈
13008:(2009).
12858:Elsevier
12671:(2009).
12604:See also
9443:feasible
9004:, i.e.,
7269:→
7226:→
7091:, where
6195:the set
5984:′
5959:′
5346:the set
4760:for any
4458:unbiased
4411:, i.e.,
3831:for all
2714:Suppose
1977:Minimize
1479:Minimize
36:modeling
13033:2562798
12968:1315300
12696:2562798
9410:, with
6604:, then
5216:. Then
4195:By the
2788:. Such
65:Methods
55:finance
13092:
13031:
13021:
12966:
12956:
12908:
12888:
12864:
12772:
12762:
12694:
12684:
12630:FortSP
12568:EMP SP
10645:given
10537:where
7958:(here
7884:where
7259:where
6097:where
4456:is an
3720:, and
1372:, and
1135:, say
1000:linear
515:where
210:where
13040:(PDF)
13015:(PDF)
12822:(PDF)
12703:(PDF)
12678:(PDF)
12586:SAMPL
12578:and
12562:AIMMS
7016:, of
6747:then
5674:then
3596:Here
1370:CPLEX
1337:When
1034:known
13090:ISBN
13068:help
13019:ISBN
12954:ISBN
12906:ISBN
12886:ISBN
12862:ISBN
12770:ISSN
12760:ISBN
12731:help
12682:ISBN
12590:AMPL
12572:GAMS
12496:for
12178:and
11694:for
10717:and
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23:.
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