Application of the Enhanced Semidefinite Relaxation Method to Construction of the Optimal Anisotropy Function

TL;DR

This paper introduces an enhanced semidefinite relaxation method to solve non-convex quadratic optimization problems, specifically applied to constructing optimal anisotropy functions for complex shapes like snowflakes.

Contribution

It develops a novel enhancement to semidefinite relaxation techniques by adding quadratic-linear constraints, improving solutions for anisotropy function construction.

Findings

Method successfully constructs optimal anisotropy functions for complex shapes.

Computational examples include real snowflake boundaries.

Enhanced relaxation guarantees solution optimality under certain conditions.

Abstract

In this paper we propose and apply the enhanced semidefinite relaxation technique for solving a class of non-convex quadratic optimization problems. The approach is based on enhancing the semidefinite relaxation methodology by complementing linear equality constraints by quadratic-linear constrains. We give sufficient conditions guaranteeing that the optimal values of the primal and enhanced semidefinite relaxed problems coincide. We apply this approach to the problem of resolving the optimal anisotropy function. The idea is to construct an optimal anisotropy function as a minimizer for the anisotropic interface energy functional for a given Jordan curve in the plane. We present computational examples of resolving the optimal anisotropy function. The examples include boundaries of real snowflakes.