[Paper Review] Tractable Subcones and LP-based Algorithms for Testing Copositivity

Akihiro Tanaka, Akiko Yoshise|arXiv (Cornell University)|Jan 1, 2015

Matrix Theory and Algorithms37 references3 citations

TL;DR

This paper introduces two new, larger subcones of the copositive cone using a semidefinite basis (SD basis), enabling copositivity testing via linear programs (LPs) with O(n²) variables and O(n²) constraints. The proposed subcones retain desirable tractability properties and demonstrate improved numerical performance, making them promising for practical copositivity detection.

ABSTRACT

The authors in a previous paper devised certain subcones of the copositive cone and showed that one can detect whether a given matrix belongs to each of them by solving linear optimization problems (LPs) with O(n) variables and O(n 2 ) constraints. They also devised LP-based algorithms for testing copositivity using the subcones. In this paper, they investigate the properties of the subcones in more detail and explore better subcones of the copositive cone having desirable properties.They introduce a semidenite basis (SD basis) that is a basis of the space of n n symmetric matrices consisting of n(n + 1)=2 symmetric semidefinite matrices. Using the SD basis, they devise two new subcones for which detection can be done by solving LPs with O(n 2 ) variables and O(n 2 ) constraints. The new subcones are larger than the ones in the previous paper and inherit their nice properties. The authors also examine the efficiency of those subcones in numerical experiments. The results show that the subcones are promising for testing copositivity.

Motivation & Objective

• To develop larger and more tractable subcones of the copositive cone that preserve the property of being testable via linear programming (LP).
• To improve upon previous subcones by enhancing their size and coverage of the copositive cone while maintaining computational efficiency.
• To introduce a semidefinite basis (SD basis) as a structural tool for constructing new subcones with favorable mathematical properties.
• To evaluate the numerical efficiency and practical viability of the new subcones in testing matrix copositivity.

Proposed method

• Constructing a semidefinite basis (SD basis) consisting of n(n+1)/2 symmetric semidefinite matrices that span the space of n×n symmetric matrices.
• Using the SD basis to define two new subcones of the copositive cone that are larger than those in prior work.
• Designing LP-based algorithms to test membership in the new subcones, with O(n²) variables and O(n²) constraints.
• Formulating the detection of subcone membership as a linear optimization problem, leveraging the structure of the SD basis.
• Ensuring the new subcones inherit the desirable tractability and closure properties of earlier subcones.
• Conducting numerical experiments to compare the performance and coverage of the new subcones against previous approaches.

Experimental results

Research questions

• RQ1 Can a larger and more comprehensive subcone of the copositive cone be constructed while preserving tractability via LPs?
• RQ2 How does the use of a semidefinite basis (SD basis) enable the construction of improved subcones with better coverage of the copositive cone?
• RQ3 What is the computational complexity of testing membership in the new subcones, and how does it scale with matrix size n?
• RQ4 How do the new subcones compare in numerical performance and coverage to previously proposed subcones?
• RQ5 Can the new subcones be efficiently implemented in practice for real-world copositivity testing?

Key findings

• The proposed subcones are strictly larger than those introduced in the authors' previous work, offering improved coverage of the copositive cone.
• Membership in the new subcones can be tested using linear programs with O(n²) variables and O(n²) constraints, maintaining computational tractability.
• The subcones inherit the favorable properties of earlier subcones, such as closure under nonnegative scaling and preservation of copositivity under certain transformations.
• Numerical experiments confirm that the new subcones are effective and efficient for testing copositivity, demonstrating practical promise.
• The use of the semidefinite basis (SD basis) enables a systematic and structured construction of subcones with enhanced theoretical and computational properties.