Convergence of the Lasserre Hierarchy of SDP Relaxations for Convex Polynomial Programs without Compactness

TL;DR

This paper proves that the Lasserre hierarchy of SDP relaxations converges for convex polynomial programs even without compactness, using coercivity and saddle-point conditions, and establishes finite convergence criteria.

Contribution

It extends the convergence analysis of the Lasserre hierarchy to non-compact convex polynomial problems by leveraging coercivity and saddle-point conditions.

Findings

Asymptotic convergence of the hierarchy without compactness.

Finite convergence guaranteed by saddle-point positive definiteness.

New sum-of-squares representation for convex polynomials.

Abstract

The Lasserre hierarchy of semidefinite programming (SDP) relaxations is an effective scheme for finding computationally feasible SDP approximations of polynomial optimization over compact semi-algebraic sets. In this paper, we show that, for convex polynomial optimization, the Lasserre hierarchy with a slightly extended quadratic module always converges asymptotically even in the face of non-compact semi-algebraic feasible sets. We do this by exploiting a coercivity property of convex polynomials that are bounded below. We further establish that the positive definiteness of the Hessian of the associated Lagrangian at a saddle-point (rather than the objective function at each minimizer) guarantees finite convergence of the hierarchy. We obtain finite convergence by first establishing a new sum-of-squares polynomial representation of convex polynomials over convex semi-algebraic sets under a saddle-point condition. We finally prove that the existence of a saddle-point of the Lagrangian for a convex polynomial program is also necessary for the hierarchy to have finite convergence.