Sparse Conic Reformulation of Structured QCQPs based on Copositive Optimization with Applications in Stochastic Optimization

TL;DR

This paper introduces a novel convex reformulation for stochastic QCQPs using a generalized copositive cone, enabling tighter bounds and potential exactness in sparse optimization problems.

Contribution

It proposes a new conic reformulation based on a generalized set-completely positive cone, improving upon existing methods by providing better bounds and certificates of optimality.

Findings

Inner and outer approximations can close the optimality gap.

Numerical experiments show the effectiveness of the approximations.

Potential to achieve exact solutions outside traditional conditions.

Abstract

In an effort to develop an alternative approach to traditional sparse reformulations, we will provide a new type of convex reformulation of a large class of stochastic quadratically constrained quadratic optimization problems that is similar to Burer's reformulation, but lifts the variables into a comparatively lower dimensional space. The reformulation rests on a generalization of the set-completely positive matrix cone. This cone can then be approximated via inner and outer approximations in order to obtain upper and lower bounds, which potentially close the optimality gap, and hence can give a certificate of exactness for these sparse reformulations outside of traditional, known sufficient conditions. Finally, we provide some numerical experiments, where we asses the quality of the inner and outer approximations, thereby showing that the approximations may indeed close the optimality gap in interesting cases.