Convex Relaxation of Bilinear Matrix Inequalities Part I: Theoretical Results

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.