Solution to the inverse Wulff problem by means of the enhanced semidefinite relaxation method

Abstract

We propose a novel method of resolving the optimal anisotropy function. The idea is to construct the optimal anisotropy function as a solution to the inverse Wul problem, i.e. as a minimizer for the anisoperimetric ratio for a given Jordan curve in the plane. It leads to a nonconvex quadratic optimization problem with linear matrix inequalities. In order to solve it we propose the so-called enhanced semide nite relaxation method which is based on a solution to a convex semide nite problem obtained by a semide nite relaxation of the original problem augmented by quadratic-linear constraints. We show that the sequence of nite-dimensional approximations of the optimal anisoperimetric ratio converges to the optimal anisoperimetric ratio which is a solution to the inverse Wul problem. Several computational examples, including those corresponding to boundaries of real snow akes, and discussion on the rate of convergence of numerical method are also presented in this paper.

On construction of upper and lower bounds for the HOMO-LUMO spectral gap

“…We refer the reader to the book [4] by Boyd and Vanderberghe on recent developments on semidefinite relaxation methods for solving nonconvex and mixed integer nonlinear optimization problems. In general, semidefinite relaxations of an original nonconvex problem can be constructed by means of the second Lagrangian dual problem which is already a convex semidefinite problem (see e.g.Ševčovič and Trnovská [25]).…”

“…In general, semidefinite relaxations of an original nonconvex problem can be constructed by means of the second Lagrangian dual problem which is already a convex semidefinite problem (see e.g. Ševčovič and Trnovská [25]).…”