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

Mitsuhiro Nishijima, Pierre-Louis Poirion, Akiko Takeda

TL;DR

This work reformulates copositive programming as a convex semi-infinite program and introduces an inexact subgradient method that provides a non-asymptotic convergence guarantee even when subproblems are solved approximately. The subproblem at each iteration reduces to a standard quadratic program, which the authors solve exactly via MILP or inexactly via deterministic grid discretization and randomized sampling on the simplex or its regular grid. They establish explicit iteration bounds and discuss how to estimate Lipschitz constants and maximize-subproblem approximations $G(x)$, enabling robust performance across problem scales. The framework is demonstrated on testing complete positivity of matrices and extended to COPP over symmetric cones, with comprehensive numerical experiments showing when each subproblem method is preferable. Overall, the paper delivers a practically viable, theoretically supported approach for large-scale copositive problems that previously lacked non-asymptotic guarantees for inexact subproblem solutions.

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 $ε> 0$ for both the objective function and the constraint arising from the copositivity condition, the proposed algorithm yields an approximate solution after $O(ε^{-2})$ 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.

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

TL;DR

This work reformulates copositive programming as a convex semi-infinite program and introduces an inexact subgradient method that provides a non-asymptotic convergence guarantee even when subproblems are solved approximately. The subproblem at each iteration reduces to a standard quadratic program, which the authors solve exactly via MILP or inexactly via deterministic grid discretization and randomized sampling on the simplex or its regular grid. They establish explicit iteration bounds and discuss how to estimate Lipschitz constants and maximize-subproblem approximations , enabling robust performance across problem scales. The framework is demonstrated on testing complete positivity of matrices and extended to COPP over symmetric cones, with comprehensive numerical experiments showing when each subproblem method is preferable. Overall, the paper delivers a practically viable, theoretically supported approach for large-scale copositive problems that previously lacked non-asymptotic guarantees for inexact subproblem solutions.

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 for both the objective function and the constraint arising from the copositivity condition, the proposed algorithm yields an approximate solution after 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.

Paper Structure

This paper contains 16 sections, 8 theorems, 83 equations, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An inexact subgradient algorithm for convex semi-infinite programming problems
  4. Application to copositive programming problems
  5. General copositive programming problems
  6. Exact method through mixed-integer linear programming
  7. Inexact deterministic method through a regular grid of the standard simplex
  8. Inexact randomized method through uniform sampling from the standard simplex
  9. Inexact randomized method through uniform sampling from a regular grid of the standard simplex
  10. Testing complete positivity of a matrix
  11. The case of symmetric cones beyond nonnegative orthants
  12. Numerical experiments
  13. Comparison among the methods for solving standard quadratic programming problems
  14. Solving general copositive programming problems
  15. Testing for non-complete positivity of exceptional doubly nonnegative matrices
  16. ...and 1 more sections

Key Result

Lemma 3.3

Suppose that Assumption asm:SIP holds. Let $\epsilon > 0$ and $N$ be a positive integer such that Then the set $I_N$ defined in eq:def_IN is non-empty, and for a minimizer $k^*$ of $f(x_k)$ over $k \in I_N$, the following two inequalities hold:

Theorems & Definitions (20)

  • Remark 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • proof
  • Remark 4.3
  • Lemma 4.4
  • ...and 10 more