Lossless Convexification and Duality

TL;DR

This paper explores conditions under which certain non-convex semidefinite programming problems exhibit strong duality, extending classical convex optimization results to a specific non-convex setting relevant in control applications.

Contribution

It proves that a class of structured non-convex SDPs satisfies strong duality under Slater's condition, which was previously known mainly for convex problems.

Findings

Non-convex SDPs with special structures satisfy strong duality.

Examples demonstrate the applicability of the theoretical results.

Results deepen understanding of non-convex SDPs in control design.

Abstract

The main goal of this paper is to investigate strong duality of non-convex semidefinite programming problems (SDPs). In the optimization community, it is well-known that a convex optimization problem satisfies strong duality if the Slater's condition holds. However, this result cannot be directly generalized to non-convex problems. In this paper, we prove that a class of non-convex SDPs with special structures satisfies strong duality under the Slater's condition. Such a class of SDPs arises in SDP-based control analysis and design approaches. Throughout the paper, several examples are given to support the proposed results. We expect that the proposed analysis can potentially deepen our understanding of non-convex SDPs arising in the control community, and promote their analysis based on KKT conditions.