Convex Relaxation of Bilinear Matrix Inequalities Part I: Theoretical Results

TL;DR

This paper introduces a new family of convex relaxations, including a computationally efficient parabolic relaxation, for solving bilinear matrix inequality problems, along with penalty functions to recover feasible solutions.

Contribution

It proposes a novel parabolic relaxation and penalty methods that improve the efficiency and feasibility recovery in BMI optimization compared to existing SDP and SOCP relaxations.

Findings

Parabolic relaxation is computationally efficient and relies only on quadratic constraints.

Penalty functions can recover feasible points if initial points are close to the feasible set.

Demonstrated effectiveness on benchmark control synthesis problems.

Abstract

This two-part paper is concerned with the problem of minimizing a linear objective function subject to a bilinear matrix inequality (BMI) constraint. In this part, we first consider a family of convex relaxations which transform BMI optimization problems into polynomial-time solvable surrogates. As an alternative to the state-of-the-art semidefinite programming (SDP) and second-order cone programming (SOCP) relaxations, a computationally efficient parabolic relaxation is developed, which relies on convex quadratic constraints only. Next, we developed a family of penalty functions, which can be incorporated into the objective of SDP, SOCP, and parabolic relaxations to facilitate the recovery of feasible points for the original non-convex BMI optimization. Penalty terms can be constructed using any arbitrary initial point. We prove that if the initial point is sufficiently close to the feasible set, then the penalized relaxations are guaranteed to produce feasible points for the original BMI. In Part II of the paper, the efficacy of the proposed penalized convex relaxations is demonstrated on benchmark instances of H2 and Hinf optimal control synthesis problems.