Parabolic Relaxation for Quadratically-constrained Quadratic Programming -- Part II: Theoretical & Computational Results

TL;DR

This paper presents theoretical convergence guarantees and computational results for a convex parabolic relaxation method applied to quadratically-constrained quadratic programming, demonstrating its effectiveness on benchmark and large-scale problems.

Contribution

It provides a convergence analysis of the sequential penalized parabolic relaxation algorithm and validates its efficiency through numerical experiments.

Findings

Algorithm converges to KKT points from feasible solutions.

Effective on benchmark non-convex QCQP problems.

Performs well on large-scale system identification instances.

Abstract

In the first part of this work [32], we introduce a convex parabolic relaxation for quadratically-constrained quadratic programs, along with a sequential penalized parabolic relaxation algorithm to recover near-optimal feasible solutions. In this second part, we show that starting from a feasible solution or a near-feasible solution satisfying certain regularity conditions, the sequential penalized parabolic relaxation algorithm convergences to a point which satisfies Karush-Kuhn-Tucker optimality conditions. Next, we present numerical experiments on benchmark non-convex QCQP problems as well as large-scale instances of system identification problem demonstrating the efficiency of the proposed approach.