A Survey of Hidden Convex Optimization

Abstract

Motivated by the fact that not all nonconvex optimization problems are difficult to solve, we survey in this paper three widely-used ways to reveal the hidden convex structure for different classes of nonconvex optimization problems. Finally, ten open problems are raised.Keywords convex programming · quadratic programming · quadratic matrix programming · fractional programming · Lagrangian dual · semidefinite programming Mathematics Subject Classification (2010) 90C20, 90C25, 90C26, 90C32

Polyak's convexity theorem, Yuan's lemma and S-lemma: extensions and applications

“…By (17), it holds that t i ≥ 0 for i = 0, 1, • • • , m. If t 0 = 0, taking x = x in (16) yields that t i = 0, i = 1, • • • , m, which contradicts the assumption t = 0. Thus, we have t 0 > 0.…”

“…S-lemma and its variants play a great role in revealing the hidden convexity of quadratic optimization problems, see [17] and references therein. As applications of Theorems 10 and 11, we study the homogeneous quadratic optimization in R n (n ≥ 3) with two bilateral quadratic form constraints:…”

Trust-region and $p$-regularized subproblems: local nonglobal minimum is the second smallest objective function value among all first-order stationary points

“…Nonconvex (TRS) has the property of zero Lagrangian-duality gap, see the recent survey on hidden convexity [4]. There is a necessary and sufficient optimality condition for the global minimizer, established in the early 1980s [5,6,7].…”

Conditional Infimum and Hidden Convexity in Optimization

“…The vocable of "hidden convexity" covers different approaches: duality and biduality analysis like in [6]; identifying classes of nonconvex optimization problems whose convex relaxations have optimal solutions which at the same time are global optimal solutions of the original nonconvex problems [5]. A survey of hidden convex optimization can be found in [15], with its focus on three widely used ways to reveal the hidden convex structure for different classes of nonconvex optimization problems. In this paper, we propose a definition of "hidden convexity" and a new way to reveal it by means of what we call the "conditional infimum".…”