43:
MMA was introduced by
Krister Svanberg in a 1987 paper titled, "The method of moving asymptotes—a new method for structural optimization." The method was proposed as an alternative to traditional optimization methods, offering an approach that could handle large-scale problems, especially in the
44:
realm of structural design. Another paper was published in 1993 by
Svanberg which added some extensions to the method, including mini-max formulations and first and second order dual methods to solve subproblems. Another version that is globally convergent was proposed by Zillober.
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The moving asymptotes serve as an adaptive mechanism. They shift and change with each iteration, progressively closing in on the optimal solution. This ensures that the approximations become increasingly accurate as the algorithm progresses.
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and objective function with a simpler, convex approximation. This approximation is represented by linear constraints and a convex objective function.
688:
615:
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The Method of Moving
Asymptotes functions as an iterative scheme. The key idea behind MMA is to approximate the original
532:
153:
510:: Design of aircraft wings and other components to reduce drag, weight, and ensure structural integrity.
24:
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28:
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8:
665:
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The Method of Moving
Asymptotes has been widely applied in various fields including:
669:
649:
603:
516:: Topology optimization for designing materials with desired mechanical properties.
522:: Optimization of machine parts for weight reduction, durability, and performance.
607:
59:
Starting from an initial guess, each iteration consists of the following steps:
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682:
661:
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and let the optimal solution of this subproblem be the next iteration point
27:
developed by
Krister Svanberg in the 1980s. It's primarily used for solving
578:
The method of moving asymptotes—a new method for structural optimization
653:
602:, NATO ASI Series, Dordrecht: Springer Netherlands, pp. 555–566,
637:
638:"A globally convergent version of the method of moving asymptotes"
31:
problems, particularly those related to structural design and
596:"The Method of Moving Asymptotes (MMA) with Some Extensions"
582:
International
Journal for Numerical Methods in Engineering
560:
Topology optimization: theory, methods, and applications
504:: Design of truss structures, beams, plates, and shells.
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594:Svanberg, K. (1993), Rozvany, G. I. N. (ed.),
558:Bendsøe, M. P., & Sigmund, O. (2003).
635:
600:Optimization of Large Structural Systems
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372:, based on the calculations from Step I.
484:and return to Step I until convergence.
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195:{\displaystyle \nabla f_{i}(x^{(k)})}
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572:
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334:by approximating explicit functions
689:Optimization algorithms and methods
307:, the (usually implicit) functions
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533:Sequential quadratic programming
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239:{\displaystyle i=0,1,\dots ,m}
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143:{\displaystyle f_{i}(x^{(k)})}
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1:
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608:10.1007/978-94-010-9577-8_26
7:
636:Zillober, C. (1993-09-01).
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520:Mechanical component design
365:{\displaystyle f_{i}^{(k)}}
21:Method of Moving Asymptotes
10:
705:
38:
445:{\displaystyle x^{(k+1)}}
68:Given an iteration point
508:Aeroelastic optimization
642:Structural Optimization
502:Structural optimization
406:{\displaystyle P^{(k)}}
280:{\displaystyle P^{(k)}}
94:{\displaystyle x^{(k)}}
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254:Generate a subproblem
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54:non-linear constraints
29:non-linear programming
25:optimization algorithm
16:Optimization algorithm
576:Svanberg, K. (1987).
538:Topology optimization
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477:{\displaystyle k=k+1}
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408:
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327:{\displaystyle f_{i}}
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33:topology optimization
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562:. Berlin: Springer.
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654:10.1007/BF01743509
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150:and the gradients
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48:Algorithm overview
617:978-94-010-9577-8
584:, 24(2), 359-373.
300:{\displaystyle P}
287:by replacing, in
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514:Material design
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648:(3): 166–174.
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621:, retrieved
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496:
493:Applications
487:
375:
249:
101:, calculate
63:
58:
51:
42:
23:(MMA) is an
20:
18:
623:2023-09-01
544:References
662:1615-1488
228:…
158:∇
683:Category
670:54187414
527:See also
376:Step III
250:Step II
39:History
668:
660:
614:
452:. Let
380:Solve
64:Step I
666:S2CID
658:ISSN
612:ISBN
202:for
19:The
650:doi
604:doi
685::
664:.
656:.
644:.
640:.
610:,
598:,
580:.
567:^
551:^
35:.
672:.
652::
646:6
606::
472:1
469:+
466:k
463:=
460:k
438:)
435:1
432:+
429:k
426:(
422:x
399:)
396:k
393:(
389:P
358:)
355:k
352:(
347:i
343:f
320:i
316:f
295:P
273:)
270:k
267:(
263:P
246:.
234:m
231:,
225:,
222:1
219:,
216:0
213:=
210:i
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185:)
182:k
179:(
175:x
171:(
166:i
162:f
138:)
133:)
130:k
127:(
123:x
119:(
114:i
110:f
87:)
84:k
81:(
77:x
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