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A method for pointwise evaluation of polyconvex envelopes

Abstract:

We investigate a method for computing the value of a polyconvex envelope at a given feed A0 ∈ ℝm×n. The method generalizes an approach for computing convex envelopes proposed by Michelsen (Fluid Phase Equilibria 9:1-19, 1982; Fluid Phase Equilibria 9:21-40, 1982) and later implemented by McKinnon and Mongeau (J Glob Optim 12(4):325-351, 1998). We formulate the problem as a primal-dual nonlinear optimization task in p(1 + mn) variables Λ,A) ∈ ℝp × (Rm×n)p subject to τ + 1) ≡ binom(m + n, n) equality constraints; and we prove that under reasonable assumptions, the global minimum of the dual problem is attained so that as a consequence, the polyconvex envelope value can be computed pointwise. A similar alternating procedure based on linear programming and adaptive mesh refinements was investigated in Bartels (SIAM J Numer Anal 43(1):363-385, 2005). The underlying function E is assumed to be at least lower semicontinuous and coercive for the existence theory. For the algorithm we require continuous differentiability. © 2011 African Mathematical Union and Springer-Verlag.