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Computational Issues for Optimal Shape Design in Hemodynamics Read chapter Read first chapter

2016 | OriginalPaper | Chapter

Proximal Bundle Method for Nonsmooth and Nonconvex Multiobjective Optimization

Authors : Marko M. Mäkelä, Napsu Karmitsa, Outi Wilppu

Published in: Mathematical Modeling and Optimization of Complex Structures

Publisher: Springer International Publishing

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Abstract

We present a proximal bundle method for finding weakly Pareto optimal solutions to constrained nonsmooth programming problems with multiple objectives. The method is a generalization of proximal bundle approach for single objective optimization. The multiple objective functions are treated individually without employing any scalarization. The method is globally convergent and capable of handling several nonconvex locally Lipschitz continuous objective functions subject to nonlinear (possibly nondifferentiable) constraints. Under some generalized convexity assumptions, we prove that the method finds globally weakly Pareto optimal solutions. Concluding, some numerical examples illustrate the properties and applicability of the method.

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previous chapter On Bifurcation Analysis of Implicitly Given Functionals in the Theory of Elastic Stability
next chapter Efficient Parallel Nash Genetic Algorithm for Solving Inverse Problems in Structural Engineering
Literature
1.
2.
N.V. Banichuk, P. Neittaanmäki, Structural Optimization with Uncertainties, vol. 162, Solid Mechanics and Its Applications (Springer, Berlin, 2010)
3.
F.H. Clarke, Optimization and Nonsmooth Analysis (Wiley, New York, 1983)MATH
4.
M. Gaudioso, M.F. Monaco, Quadratic approximations in convex nondifferentiable optimization. SIAM J. Control Optim. 29(1), 58–70 (1991)MathSciNetCrossRefMATH
5.
J. Haslinger, P. Neittaanmäki, Finite Element Approximation for Optimal Shape, Material and Topology Design (Wiley, Chichester, 1996)MATH
6.
J.-B. Hiriart-Urruty, New concepts in nondifferentiable programming. Bull. Soc. Math. France Mém 60, 57–85 (1979)MathSciNetMATH
7.
K.C. Kiwiel, A descent method for nonsmooth convex multiobjective minimization. Large Scale Syst. 8(2), 119–129 (1985)MathSciNetMATH
8.
K.C. Kiwiel, Methods of Descent for Nondifferentiable Optimization, vol. 1133, Lecture Notes in Mathematics (Springer, Berlin, 1985)
9.
10.
C. Lemaréchal. Nondifferentiable Optimization, eds. by G.L. Nemhauser, A.H.G.Rinnooy Kan, M.J. Todd. Optimization (North-Holland, Amsterdam, 1989), pp. 529–572
11.
L. Lukšan, J. Vlček, Globally convergent variable metric method for convex nonsmooth unconstrained minimization. J. Optim. Theory Appl. 102(3), 593–613 (1999)MathSciNetCrossRefMATH
12.
13.
M.M. Mäkelä, P. Neittaanmäki, Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control (World Scientific, Singapore, 1992)CrossRefMATH
14.
M.M. Mäkelä, V.-P. Eronen, N. Karmitsa, On nonsmooth optimality conditions with generalized convexities. TUCS Technical Reports 1056, Turku Centre for Computer Science, Turku, 2012
15.
M.M. Mäkelä, V.-P. Eronen, N. Karmitsa, On nonsmooth multiobjective optimality conditions with generalized convexities, in Optimization in Science and Engineering, ed. by ThM Rassias, C.A. Floudas, S. Butenko (Springer, New York, 2014), pp. 333–357CrossRef
16.
K. Miettinen, Nonlinear Multiobjective Optimization (Kluwer, Boston, 1999)MATH
17.
K. Miettinen, M.M. Mäkelä, Interactive bundle-based method for nondifferentiable multiobjective optimization: NIMBUS. Optimization 34(3), 231–246 (1995)MathSciNetCrossRefMATH
18.
R. Mifflin, A modification and an extension of Lemarechal’s algorithm for nonsmooth minimization. Math. Program. Stud. 17, 77–90 (1982)MathSciNetCrossRefMATH
19.
J.J. Moreau, P.D. Panagiotopoulos, G. Strang (eds.), Topics in Nonsmooth Mechanics (Birkhäuser, Basel, 1988)MATH
20.
21.
J. Outrata, M. Kočvara, J. Zowe, Nonsmooth approach to optimization problems with equilibrium constraints. Theory, Applications and Numerical Results (Kluwer, Dordrecht, 1998)
22.
H. Schramm, J. Zowe, A version of the bundle idea for minimizing a nonsmooth functions: conceptual idea, convergence analysis, numerical results. SIAM J. Optim. 2, 121–152 (1992)MathSciNetCrossRefMATH
23.
R.E. Steuer, Multiple Criteria Optimization: Theory, Computation, and Application (Wiley, New York, 1986)MATH
24.
S. Wang, Algorithms for multiobjective and nonsmooth optimization. In Methods of Operations Research 58 (Athenäum, Frankfurt am Main, 1989), pp. 131–142
Metadata
Title
Proximal Bundle Method for Nonsmooth and Nonconvex Multiobjective Optimization
Authors
Marko M. Mäkelä
Napsu Karmitsa
Outi Wilppu
Copyright Year
2016
Book
Mathematical Modeling and Optimization of Complex Structures

Print ISBN: 978-3-319-23563-9

Electronic ISBN: 978-3-319-23564-6

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-23564-6_12

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