Outer Approximation Scheme for Weakly Convex Constrained Optimization Problems

TL;DR

This paper introduces a novel outer approximation method using quadratic cuts for weakly convex constrained optimization, proving convergence to global minimizers and demonstrating effectiveness on packing and classification problems.

Contribution

The paper presents a new outer approximation scheme with quadratic cuts for weakly convex problems, including convergence proof and practical variants.

Findings

Convergent subsequences lead to global minimizers.

Method successfully applied to packing and classification problems.

Quadratic cuts improve approximation over linear methods.

Abstract

Outer approximation methods have long been employed to tackle a variety of optimization problems, including linear programming, in the 1960s, and continue to be effective for solving variational inequalities, general convex problems, as well as mixed-integer linear, and nonlinear programming problems. In this work, we introduce a novel outer approximation scheme specifically designed for solving weakly convex constrained optimization problems. The key idea lies in utilizing quadratic cuts, rather than the traditional linear cuts, and solving an outer approximation problem at each iteration in the form of a Quadratically Constrained Quadratic Programming (QCQP) problem. The primary result demonstrated in this work is that every convergent subsequence generated by the proposed outer approximation scheme converges to a global minimizer of the general weakly convex optimization problem under consideration. To enhance the practical implementation of this method, we also propose two variants of the algorithm. The efficacy of our approach is illustrated through its application to two distinct problems: the circular packing problem and the Neyman-Pearson classification problem, both of which are reformulated within the framework of weakly convex constrained optimization.