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Numerical optimization and quasiconvexity

Published online by Cambridge University Press:  26 September 2008

P. A. Gremaud
P. A. Gremaud
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
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Abstract

In the Calculus of Variations, several notions of convexity have emerged, corresponding to different properties of the functionals to be minimized. The relations between these various notions are not yet fully understood. In this context, we present a numerical study of quasiconvexity for some functions of the type f(ξ) = g(|ξ|2, det ξ), where Ξ is a 2×2-matrix. The corresponding global optimization problems are solved using a simulated annealing-like algorithm. The computations strongly indicate that the considered functions are quasiconvex if and only if they are rank-one convex. The relation to Morrey's conjecture, various applications and implementation problems are discussed.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 6 , Issue 1 , February 1995 , pp. 69 - 82
Copyright
Copyright © Cambridge University Press 1995

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References

[AD]Alibert, J.-J. & Dacorogna, B. 1992 An example of a quasiconvex function that is not polyconvex in two dimensions. Arch. Rat. Mech. Anal. 117, 155–166.CrossRefGoogle Scholar
[APZ]Aluffi-Pentini, F., Parisi, V. & Zirilli, F. 1985 Global optimization and stochastic differential equations. J. Optim. Theory Appl. 47, 1–16.CrossRefGoogle Scholar
[BC]Brighi, B. & Chipot, M.Approximated convex envelope of a function. Preprint.Google Scholar
[CCK]Chipot, M., Collins, C. & Kinderlehrer, D.Numerical analysis of oscillations in multiple well problems. In preparation.Google Scholar
[CHS]Chiang, , Tzuu-Shuh, , Hwang, , Chii-Ruey, & Sheu, , Shuenn-Jyi, 1987 Diffusion For Global Optimization In RN. Siam J. Control And Optimization 25, 737–753.CrossRefGoogle Scholar
[CL]Collins, C. & Luskin, M. 1991 Optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem. Math. Comp. 57, 621–637.CrossRefGoogle Scholar
[CL2]Collins, C. & Luskin, M. 1989 The computation of austenitic-martensitic phase transition, in Partial Differential Equations and Continuum Models of Phase Transitions, Rascle, M., Serre, D. & Slemrod, M. (eds.), Lecture Notes in Physics, Springer-Verlag, pp. 34–50.Google Scholar
[DA]Dacorogna, B. 1989 Direct Methods in the Calculus of Variations. Springer-Verlag.CrossRefGoogle Scholar
[DDGR]Dacorogna, B., Douchet, J., Gangbo, W. & Rappaz, J. 1990 Some examples of rank one convex functions in dimension two. Proc. Royal Soc. Edinburgh 114A, 135–150.CrossRefGoogle Scholar
[FR]Friedman, A. 1975 Stochastic Differential Equations and Applications, vol. 1. Academic Press.Google Scholar
[GI]Gidas, B. 1991 Metropolis-type Monte Carlo Simulation Algorithms and Simulated Annealing. Preprint, Brown University.Google Scholar
[GM]Gelfand, S. B. & Mitter, S. K. 1991 Recursive stochastic algorithms for global optimization in Rd. SIAM J. Control and Optimization 29, 999–1018.CrossRefGoogle Scholar
[GR]Gremaud, P. A. 1992 Numerical Analysis of a Nonconvex Variational Problem Related to Solid-Solid Phase Transitions. IMA Preprint Series # 1001. (To be published in SIAM J. Numer. Anal.)Google Scholar
[KS]Kohn, R. V. & Strang, G. 1986 Optimal design and relaxation of variational problems. Comm. Pure Appl. Math. 39.CrossRefGoogle Scholar
[LM]Luskin, M. & MA, LING. Numerical Optimization of the Micromagnetics Energy. Preprint.Google Scholar
[MO]Morrey, C. B. 1952 Quasiconvexity and the semicontinuity of multiple integrals. Pacific J. Math. 2, 25–53.CrossRefGoogle Scholar
[RI]Ripley, B. D. 1990 Thoughts on pseudorandom number generators. J. Comput. Appl. Math. 31, 153–163.CrossRefGoogle Scholar
[SC]Schuss, Z. 1980 Theory and Applications of Stochastic Differential Equations. Wiley.Google Scholar
[SV] ŠveràK, V. 1992 Rank one convexity does not imply quasiconvexity. Proc. Royal Soc. Edinburgh 120A, 185–189.Google Scholar