Inexact subgradient algorithm with a non-asymptotic convergence guarantee for copositive programming problems

TL;DR

This paper introduces a subgradient algorithm with non-asymptotic guarantees for copositive programming, capable of handling inexact subproblem solutions and applicable to testing matrix complete positivity.

Contribution

It presents a novel subgradient method with explicit iteration bounds for copositive problems, including inexact subproblem solutions, and applies it to matrix positivity testing.

Findings

Algorithm achieves approximate solutions within $O( ext{ extbackslash epsilon}^{-2})$ iterations.

Numerical experiments compare exact and inexact quadratic programming approaches.

Method effectively detects non-complete positivity in various matrices.

Abstract

In this paper, we propose a subgradient algorithm with a non-asymptotic convergence guarantee to solve copositive programming problems. The subproblem to be solved at each iteration is a standard quadratic programming problem, which is NP-hard in general. However, the proposed algorithm allows this subproblem to be solved inexactly. For a prescribed accuracy $\epsilon > 0$ for both the objective function and the constraint arising from the copositivity condition, the proposed algorithm yields an approximate solution after $O(\epsilon^{-2})$ iterations, even when the subproblems are solved inexactly. We also discuss exact and inexact approaches for solving standard quadratic programming problems and compare their performance through numerical experiments. In addition, we apply the proposed algorithm to the problem of testing complete positivity of a matrix and derive a sufficient condition for certifying that a matrix is not completely positive. Experimental results demonstrate that we can detect the lack of complete positivity in various doubly nonnegative matrices that are not completely positive.