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Numerical optimization and quasiconvexity. (English) Zbl 0824.65043
A global minimization algorithm which belongs to the class of simulated annealing methods is introduced. It is based on ideas ranging from quantum mechanics and stochastic differential equations to more traditional descent methods (gradient or conjugate gradient). It is applied to the two-dimensional minimization problem \(\min \{I(u);\;u\in W^{1, \infty}_ 0 (\Omega, \mathbb{R}^ 2)\}\), with \(I(u) = \int_ \Omega f(\nabla u(x))dx\), where \(\Omega \subset \mathbb{R}^ 2\) is an open bounded set and \(f : \mathbb{R}^{2\times 2} \to \mathbb{R}\) is a continuous function. The discretization of the problem is made by the finite element method.
The algorithm is numerically tested for functions which are considered in relation with Morrey’s conjecture (i.e. candidates for non-quasiconvex functions which are rank-one convex). The computations strongly suggest that those functions are quasiconvex if and only if they are rank-one convex, leaving thus Morrey’s conjecture unanswered in 2D, but ruling out a family of possible candidates.
Reviewer: V.Arnăutu (Iaşi)

MSC:
65K10 Numerical optimization and variational techniques
90C52 Methods of reduced gradient type
49J20 Existence theories for optimal control problems involving partial differential equations
49M15 Newton-type methods
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