Generalizations of doubly nonnegative cones and their comparison

TL;DR

This paper explores extensions of the doubly nonnegative cone to create tighter relaxations for generalized completely positive programming, comparing their theoretical strengths and demonstrating improved bounds through numerical experiments.

Contribution

It introduces two new generalized DNN cones derived from inner-approximation hierarchies and compares their relaxation strengths with existing cones for GCPP.

Findings

GDNN cones provide significantly tighter bounds than the positive semidefinite cone.

Theoretical comparisons highlight the relaxation strengths of the new GDNN cones.

Numerical results confirm the effectiveness of the proposed cones in GCPP relaxations.

Abstract

In this study, we examine the various extensions of the doubly nonnegative (DNN) cone, frequently used in completely positive programming (CPP) to achieve a tighter relaxation than the positive semidefinite cone. To provide tighter relaxation for generalized CPP (GCPP) than the positive semidefinite cone, inner-approximation hierarchies of the generalized copositive cone are exploited to obtain two generalized DNN (GDNN) cones from the DNN cone. This study conducts theoretical and numerical comparisons to assess the relaxation strengths of the two GDNN cones over the direct products of a nonnegative orthant and second-order or positive semidefinite cones. These comparisons also include an analysis of the existing GDNN cone proposed by Burer and Dong. The findings from solving several GDNN programming relaxation problems for a GCPP problem demonstrate that the three GDNN cones provide significantly tighter bounds for GCPP than the positive semidefinite cone.