Hidden Convexity in Partially Separable Optimization

Abstract

The paper identifies 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. Such a hidden convexity property was so far limited to quadratically constrained quadratic problems with one or two constraints. We extend it here to problems with some partial separable structure. Among other things, the new hidden convexity results open up the possibility to solve multi-stage robust optimization problems using certain nonlinear decision rules.

Deriving robust counterparts of nonlinear uncertain inequalities

“…In case strong duality does not hold, we still have (by weak duality) that (10) implies (9). Hence, whenever x and some b solve (11), then x satisfies (6), i.e.…”

“…Under suitable convexity and regularity conditions on f (., x) and U (such as (7)) strong duality holds between the maximization problem in (8) and (9), hence x is robust feasible if and only if…”

“…Reparametrizing and using "hidden concavity" results. For special types of constraints we can use the "hidden concavity" result of [9]. Suppose (RC) is of the following form…”

“…It has been proved in [9] that if b > f (ā), then there is another optimal solution for a that satisfies b = f (a). Hence, the RC (40) is equivalent to (41), for which Theorem 2 can be applied.…”

Conditional Infimum and Hidden Convexity in Optimization

“…This is why other approaches have been developed, like convexification by domain or range transformation as exposed in [8] which provides a survey. 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.…”