Local Optimality Conditions for a Class of Hidden Convex Optimization

TL;DR

This paper investigates local optimality conditions in hidden convex optimization problems, especially those combining trust-region subproblems with convex optimization, revealing complex local minima structures.

Contribution

It introduces a new class of hidden convex problems and provides comprehensive conditions for local optimality, highlighting unexpected multiple local minima.

Findings

Hidden convex problems can have multiple local non-global minima.

Trust-region subproblems typically have at most one local non-global minimizer.

Joint problems may have more than one local non-global minimizer despite individual properties.

Abstract

Hidden convex optimization is such a class of nonconvex optimization problems that can be globally solved in polynomial time via equivalent convex programming reformulations. In this paper, we focus on checking local optimality in hidden convex optimization. We first introduce a class of hidden convex optimization problems by jointing the classical nonconvex trust-region subproblem (TRS) with convex optimization (CO), and then present a comprehensive study on local optimality conditions. In order to guarantee the existence of a necessary and sufficient condition for local optimality, we need more restrictive assumptions. To our surprise, while (TRS) has at most one local non-global minimizer and (CO) has no local non-global minimizer, their joint problem could have more than one local non-global minimizer.