Alternative SDP and SOCP Approximations for Polynomial Optimization

TL;DR

This paper explores alternative second-order cone programming (SOCP) relaxations for polynomial optimization, demonstrating their effectiveness in improving solution quality and convergence compared to traditional SDP hierarchies, especially for large-scale problems.

Contribution

It introduces new SOCP-based hierarchies that enhance linear programming relaxations and converge to the optimal value for compact feasible sets, offering a computationally efficient alternative.

Findings

SOCP hierarchies can strengthen LP relaxations for POPs.

These hierarchies outperform SDP relaxations on certain challenging POPs.

Convergence to the optimal value is achieved when the feasible set is compact.

Abstract

In theory, hierarchies of semidefinite programming (SDP) relaxations based on sum of squares (SOS) polynomials have been shown to provide arbitrarily close approximations for a general polynomial optimization problem (POP). However, due to the computational challenge of solving SDPs, it becomes difficult to use SDP hierarchies for large-scale problems. To address this, hierarchies of second-order cone programming (SOCP) relaxations resulting from a restriction of the SOS polynomial condition have been recently proposed to approximate POPs. Here, we consider alternative ways to use this SOCP restrictions of the SOS condition. In particular, we show that SOCP hierarchies can be effectively used to strengthen hierarchies of linear programming (LP) relaxations for POPs. Specifically, we show that this solution approach is substantially more effective in finding solutions of certain POPs for which the more common hierarchies of SDP relaxations are known to perform poorly. Furthermore, when the feasible set of the POP is compact, these SOCP hierarchies converge to the POP's optimal value.