Robust SOS-Convex Polynomial Programs: Exact SDP Relaxations

TL;DR

This paper develops exact semidefinite programming relaxations for a broad class of robust convex polynomial programs, enabling efficient solutions under data uncertainty with various uncertainty sets.

Contribution

It introduces sums of squares representations and establishes exact SDP-relaxations for robust SOS-convex programs, extending to polytopic and ellipsoidal uncertainty sets.

Findings

Exact SDP-relaxations are possible for robust SOS-convex programs.

Polytopic and ellipsoidal uncertainty sets allow second-order cone reformulations.

The methods enable efficient robust optimization under common uncertainty models.

Abstract

This paper studies robust solutions and semidefinite linear programming (SDP) relaxations of a class of convex polynomial programs in the face of data uncertainty. The class of convex programs, called robust SOS-convex programs, includes robust quadratically constrained convex programs and robust separable convex polynomial programs. It establishes sums of squares polynomial representations characterizing robust solutions and exact SDP-relaxations of robust SOS-convex programs under various commonly used uncertainty sets. In particular, the results show that the polytopic and ellipsoidal uncertainty sets, that allow second-order cone re-formulations of robust quadratically constrained programs, continue to permit exact SDP-relaxations for a broad class of robust SOS-convex programs. They also yield exact second-order cone relaxation for robust quadratically constrained programs.