On the Equivalence of SDP Feasibility and a Convex Hull Relaxation for System of Quadratic Equations

TL;DR

This paper establishes an equivalence between SDP feasibility and convex hull relaxation for quadratic systems, enabling alternative algorithms like the Triangle Algorithm for solving SDP problems efficiently.

Contribution

It introduces a novel equivalence between SDP feasibility and convex hull relaxation, and develops a Triangle Algorithm-based approach as an alternative to interior-point methods.

Findings

Triangle Algorithm computes approximate least-norm solutions efficiently.

The approach applies to SDP relaxations of binary quadratic problems like MAX-CUT.

Complexity per iteration involves eigenvalue computation or trust region subproblems.

Abstract

We show {\it semidefinite programming} (SDP) feasibility problem is equivalent to solving a {\it convex hull relaxation} (CHR) for a finite system of quadratic equations. On the one hand, this offers a simple description of SDP. On the other hand, this equivalence makes it possible to describe a version of the {\it Triangle Algorithm} for SDP feasibility based on solving CHR. Specifically, the Triangle Algorithm either computes an approximation to the least-norm feasible solution of SDP, or using its {\it distance duality}, provides a separation when no solution within a prescribed norm exists. The worst-case complexity of each iteration is computing the largest eigenvalue of a symmetric matrix arising in that iteration. Alternate complexity bounds on the total number of iterations can be derived. The Triangle Algorithm thus provides an alternative to the existing interior-point algorithms for SDP feasibility and SDP optimization. In particular, based on a preliminary computational result, we can efficiently solve SDP relaxation of {\it binary quadratic} feasibility via the Triangle Algorithm. This finds application in solving SDP relaxation of MAX-CUT. We also show in the case of testing the feasibility of a system of convex quadratic inequalities, the problem is reducible to a corresponding CHR, where the worst-case complexity of each iteration via the Triangle Algorithm is solving a {\it trust region subproblem}. Gaining from these results, we discuss potential extension of CHR and the Triangle Algorithm to solving general system of polynomial equations.