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arxiv: 2510.27160 · v3 · submitted 2025-10-31 · 🧮 math.OC

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

Mitsuhiro Nishijima , Pierre-Louis Poirion , Akiko Takeda This is my paper

Pith reviewed 2026-05-18 03:40 UTC · model grok-4.3

classification 🧮 math.OC
keywords copositive programmingsubgradient algorithminexact optimizationnon-asymptotic convergencecomplete positivityquadratic programmingmatrix certification
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The pith

A subgradient algorithm solves copositive programs to prescribed accuracy ε in O(ε^{-2}) iterations even with inexact subproblems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a subgradient algorithm for solving copositive programming problems that provides a non-asymptotic convergence guarantee. It shows that an approximate solution meeting a given accuracy ε for both the objective and the copositivity constraint can be found in O(ε^{-2}) iterations. The algorithm allows the NP-hard quadratic subproblems to be solved inexactly, as long as the error is controlled. This is useful because it makes the method practical despite the hardness of the subproblems. The approach is also applied to certify that certain matrices are not completely positive.

Core claim

We propose a subgradient algorithm for copositive programming problems with a non-asymptotic convergence guarantee. 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.

What carries the argument

Inexact subgradient iteration on the dual of the copositive program, where each step solves a standard quadratic program only to a controllable accuracy and the accumulated error is controlled by the step-size schedule.

Load-bearing premise

The errors from inexactly solving the quadratic subproblems can be bounded so that their total contribution to the objective and constraint stays below the target accuracy ε throughout the run.

What would settle it

Run the algorithm on a small known copositive program whose exact optimum is available, set ε to 0.01, and check whether an ε-approximate solution appears within roughly 10,000 iterations.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper proposes an inexact subgradient algorithm for copositive programming problems. Each iteration requires solving a standard quadratic program (StQP) that can be solved inexactly. The central claim is a non-asymptotic convergence guarantee: for any prescribed ε > 0, the algorithm produces an approximate solution satisfying ε-accuracy in both the objective and the copositivity constraint violation after O(ε^{-2}) iterations. The manuscript also discusses exact and inexact methods for the StQP subproblems, derives a sufficient condition for certifying that a matrix is not completely positive, and reports numerical experiments on detecting lack of complete positivity in doubly nonnegative matrices.

Significance. If the convergence analysis rigorously controls the accumulation of inexactness errors while preserving the O(ε^{-2}) bound, the result would supply a practical, rate-guaranteed method for an important class of NP-hard problems that arise in quadratic optimization and matrix copositivity testing. The non-asymptotic guarantee, the explicit sufficient condition for non-complete positivity, and the numerical demonstration on DNN matrices constitute concrete strengths. The work would be more significant if the required per-iteration accuracy δ(ε) is shown to be achievable by standard local or relaxation-based StQP solvers.

major comments (2)
  1. [Abstract and convergence theorem (likely §3)] The non-asymptotic claim (abstract and convergence theorem) states that O(ε^{-2}) iterations suffice even with inexact StQP solves. Standard inexact subgradient analysis requires an explicit relation between the per-step tolerance δ_k and ε (typically δ_k = O(ε^2/k) or δ ≤ ε^2/T with T = O(ε^{-2})) so that the total perturbation to the objective and copositivity residual remains ≤ ε. If the theorem assumes only an abstract inexact oracle without deriving this scaling or verifying that the discussed StQP solvers (local methods, relaxations) can meet it controllably, the iteration bound does not transfer to the stated joint accuracy in objective and constraint.
  2. [§4] §4 (application to complete positivity): the sufficient condition for certifying that a matrix is not completely positive is derived from the algorithm output, but the manuscript does not quantify how the ε-approximate solution translates into a rigorous certificate (i.e., a positive lower bound on the copositivity violation). Without an explicit error propagation argument, the certification claim rests on the same inexactness control that is questioned above.
minor comments (3)
  1. [Introduction / problem formulation] Clarify the precise reformulation of the copositive program (objective and constraint) that allows the subgradient to be obtained from an StQP; the abstract refers to “the constraint arising from the copositivity condition” without an equation number.
  2. [Numerical experiments] In the numerical section, report the actual accuracy parameters (δ or tolerance) used for the inexact StQP solves and confirm they satisfy the scaling required by the convergence theorem.
  3. Minor notation: ensure consistent use of ε for the target accuracy and δ for subproblem tolerance throughout the theorems and experiments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The concerns about explicit error control in the inexact subgradient analysis and the translation of approximate solutions into rigorous certificates are well-taken. We address each point below and will incorporate clarifications and additional arguments in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract and convergence theorem (likely §3)] The non-asymptotic claim (abstract and convergence theorem) states that O(ε^{-2}) iterations suffice even with inexact StQP solves. Standard inexact subgradient analysis requires an explicit relation between the per-step tolerance δ_k and ε (typically δ_k = O(ε^2/k) or δ ≤ ε^2/T with T = O(ε^{-2})) so that the total perturbation to the objective and copositivity residual remains ≤ ε. If the theorem assumes only an abstract inexact oracle without deriving this scaling or verifying that the discussed StQP solvers (local methods, relaxations) can meet it controllably, the iteration bound does not transfer to the stated joint accuracy in objective and constraint.

    Authors: We agree that the current statement of the theorem would be strengthened by an explicit derivation of the tolerance scaling. In the revised manuscript we will insert a lemma that specifies δ_k ≤ C ε²/k (with C depending only on problem constants such as the Lipschitz modulus of the subgradient mapping) and shows that the accumulated perturbation to both the objective value and the copositivity residual is at most ε/2 after T = O(ε^{-2}) steps. The proof follows the standard telescoping argument for inexact subgradient methods but is specialized to the copositive constraint. We will also expand the discussion of StQP solvers to indicate how the required δ_k can be achieved by controlling the termination tolerance of local solvers or by using SDP relaxations whose duality gap can be driven below δ_k; a brief reference to existing complexity results for these methods will be added. revision: yes

  2. Referee: [§4] §4 (application to complete positivity): the sufficient condition for certifying that a matrix is not completely positive is derived from the algorithm output, but the manuscript does not quantify how the ε-approximate solution translates into a rigorous certificate (i.e., a positive lower bound on the copositivity violation). Without an explicit error propagation argument, the certification claim rests on the same inexactness control that is questioned above.

    Authors: We concur that a quantitative link between the ε-approximate output and a strictly positive lower bound on the violation measure is needed for a fully rigorous certificate. In the revision we will add a short proposition that, whenever the algorithm returns a point whose objective and residual are within ε of optimality and feasibility with ε smaller than half the observed violation, the true copositivity violation is bounded below by a positive constant depending on ε and the Lipschitz constant of the violation function. The argument uses a standard perturbation lemma for the inner product with the copositive cone and will be placed immediately after the statement of the sufficient condition. revision: yes

Circularity Check

0 steps flagged

Standard inexact subgradient analysis applied to copositive reformulation; no reduction to inputs or self-citation chains.

full rationale

The paper's central result is a non-asymptotic O(ε^{-2}) iteration bound for an inexact subgradient method on a reformulation of copositive programs, where each step solves an StQP subproblem to controllable accuracy. This follows directly from classical subgradient convergence theory (with summable or bounded perturbations from inexact oracles) applied to the specific objective and copositivity residual; the abstract states the guarantee holds 'even when the subproblems are solved inexactly' without invoking fitted parameters, self-definitional loops, or load-bearing self-citations. Numerical experiments and the complete-positivity application are presented as separate illustrations, not as the source of the convergence claim. No equation or theorem in the provided description reduces the bound to a renaming or to a prior result by the same authors that itself assumes the target guarantee.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, no specific free parameters, ad-hoc axioms, or invented entities are identifiable; the work relies on standard convex optimization theory for subgradient methods.

axioms (1)
  • standard math Standard properties of subgradient methods apply to nonsmooth convex problems over the copositive cone with controllable inexact subproblem solves
    The O(ε^{-2}) rate is invoked for the overall algorithm even under inexact inner solves.

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