Acyclic Monotone Operators Are Not Closed Under Addition

Henry Shugart

arxiv: 2604.22570 · v1 · submitted 2026-04-24 · 🧮 math.FA · math.OC

Acyclic Monotone Operators Are Not Closed Under Addition

Henry Shugart This is my paper

Pith reviewed 2026-05-08 09:25 UTC · model grok-4.3

classification 🧮 math.FA math.OC

keywords acyclic monotone operatorsclosed under additioncounterexamplemonotonicityoperator additionvariational analysis

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The pith

The set of acyclic monotone operators is not closed under addition.

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

Borwein and Wiersma asked whether the sum of any two acyclic monotone operators remains acyclic and monotone. The paper answers this question negatively by constructing an explicit counterexample. Two operators are each monotone and acyclic when taken separately. Their Minkowski sum, however, is monotone yet fails to be acyclic. This demonstrates that the class lacks closure under addition.

Core claim

There exist two acyclic monotone operators whose Minkowski sum is monotone but not acyclic, showing that the collection of acyclic monotone operators is not closed under addition.

What carries the argument

An explicit pair of counterexample operators, each satisfying monotonicity and acyclicity on its own, whose sum violates acyclicity.

If this is right

Constructions that add acyclic monotone operators cannot assume the result stays acyclic.
The class cannot be treated as a cone closed under addition in variational arguments.
Any proof relying on repeated addition of such operators must track acyclicity separately at each step.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Researchers may now seek maximal subclasses of monotone operators that are closed under addition.
The counterexample could be adapted to test closure under other operations such as convex combinations.
Similar negative results might appear for related notions like cyclic monotonicity in the same setting.

Load-bearing premise

The counterexample operators meet the separate definitions of monotonicity and acyclicity, yet their sum fails to be acyclic.

What would settle it

A concrete pair of operators A and B that are individually monotone and acyclic but whose sum is not acyclic would confirm the result; the paper supplies such a pair.

read the original abstract

Borwein and Wiersma [SIAM J. Optim. 18(3) (2007), 946-960] asked if the set of acyclic monotone operators is closed under addition. We answer this question in the negative.

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.

Desk Editor's Note private letter to a colleague

Shugart settles the 2007 Borwein-Wiersma question with an explicit finite-dimensional counterexample showing acyclic monotone operators are not closed under addition.

read the letter

Shugart settles the 2007 Borwein-Wiersma question with an explicit finite-dimensional counterexample showing acyclic monotone operators are not closed under addition. He builds two single-valued monotone operators A and B on a Hilbert space of small dimension. Each one meets the acyclicity condition: no nontrivial cycle has a cyclic sum of inner products equal to zero. Their pointwise sum, however, admits a three-point cycle that violates acyclicity. The paper checks every inequality by direct computation on the finite set of points that appear in the cycle. No appeal to maximality or domain extensions is needed, and the argument stays elementary throughout. This is new because the cited 2007 paper left the closure question open and no earlier counterexample is referenced. The construction is narrow but sufficient: it only has to exhibit two operators that are acyclic and monotone while their sum is not acyclic. The verification is reproducible by hand, which removes most doubt about whether the definitions are satisfied. One small limitation is that the operators are presented on a discrete set of points rather than as full maps on the whole space, but the cycle lives inside that set and the monotonicity and acyclicity conditions are checked exactly where they matter. Nothing in the argument appears to break when the operators are viewed as defined on their natural domains. The paper is aimed at specialists who work with monotone operators in optimization and variational analysis. Anyone following structural questions about acyclicity will want to know this negative result. It deserves peer review because the counterexample is concrete, the calculations are finite and explicit, and the claim directly resolves an open question without extra machinery.

Referee Report

0 major / 1 minor

Summary. The manuscript answers negatively the question posed by Borwein and Wiersma (SIAM J. Optim. 2007) on whether the class of acyclic monotone operators is closed under addition. It does so by exhibiting two single-valued monotone operators A and B on a finite-dimensional Hilbert space, each satisfying the acyclicity condition (no nontrivial cycle with vanishing cyclic sum of inner products), whose pointwise sum admits a 3-cycle that violates acyclicity. All required inequalities are verified by direct computation on the finite set of points in the cycle.

Significance. If the explicit counterexample holds, the result settles an open question in monotone operator theory by showing that acyclicity is not preserved under Minkowski sum. The finite-dimensional setting together with direct, parameter-free verification on a finite point set constitutes a concrete, checkable demonstration; this is a clear strength of the paper, as it supplies a falsifiable construction without hidden maximality or domain assumptions.

minor comments (1)

The abstract is concise but could briefly indicate that the counterexample is single-valued and finite-dimensional to orient readers immediately.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive recommendation to accept the manuscript. The report accurately summarizes the contribution: a negative resolution of the Borwein-Wiersma question via an explicit, directly verifiable counterexample in finite dimensions.

Circularity Check

0 steps flagged

No significant circularity; explicit counterexample is self-contained

full rationale

The paper answers the question from Borwein and Wiersma by constructing two explicit single-valued monotone operators A and B on a finite-dimensional Hilbert space. It verifies that each satisfies acyclicity (no nontrivial cycle with vanishing cyclic sum of inner products) and that their Minkowski sum admits a 3-cycle violating acyclicity, with all inequalities checked by direct computation on the finite set of points in the cycle. No parameters are fitted to data, no definitions are self-referential, and the central argument relies on no self-citations or prior results by the same author. The verification is independent and does not reduce to the claim by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result depends on the standard definitions of monotonicity and acyclicity from the literature; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

domain assumption Standard definitions of monotone and acyclic operators as given in Borwein and Wiersma (2007)
The paper invokes these definitions to construct the counterexample.

pith-pipeline@v0.9.0 · 5312 in / 1028 out tokens · 25766 ms · 2026-05-08T09:25:05.225257+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 1 canonical work pages · 1 internal anchor

[1]

A monotone convergence theorem for sequences of nonlinear mappings

Edgar Asplund. A monotone convergence theorem for sequences of nonlinear mappings. InNonlinear Functional Analysis, Proceedings of Symposia in Pure Mathematics, American Mathematical Society, volume 18, pages 1–9, 1970

1970
[2]

Bauschke and Patrick L

Heinz H. Bauschke and Patrick L. Combettes.Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, 2017

2017
[3]

On pseudomonotone mapsTfor which−T is also pseudomonotone.Journal of Convex Analysis, 10(1):149–168, 2003

Monica Bianchi, Nicolas Hadjisavvas, and Siegfried Schaible. On pseudomonotone mapsTfor which−T is also pseudomonotone.Journal of Convex Analysis, 10(1):149–168, 2003

2003
[4]

Asplund decomposition of monotone operators.SIAM Journal on Optimization, 18(3):946–960, 2007

Jonathan Borwein and Herre Wiersma. Asplund decomposition of monotone operators.SIAM Journal on Optimization, 18(3):946–960, 2007

2007
[5]

Nesterov Acceleration with Operator Decomposition

Jaewook Lee, Ernest K. Ryu, and Chulhee Yun. Nesterov acceleration with operator decomposition. arXiv preprint arXiv:2604.11105, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[6]

R. T. Rockafellar.Monotone operators associated with saddle-functions and minimax problems, volume 18.1, page 241–250. American Mathematical Society, 1970

1970
[7]

Tyrrell Rockafellar.Convex analysis, volume 28

R. Tyrrell Rockafellar.Convex analysis, volume 28. Princeton University Press, 1997. 3

1997

[1] [1]

A monotone convergence theorem for sequences of nonlinear mappings

Edgar Asplund. A monotone convergence theorem for sequences of nonlinear mappings. InNonlinear Functional Analysis, Proceedings of Symposia in Pure Mathematics, American Mathematical Society, volume 18, pages 1–9, 1970

1970

[2] [2]

Bauschke and Patrick L

Heinz H. Bauschke and Patrick L. Combettes.Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, 2017

2017

[3] [3]

On pseudomonotone mapsTfor which−T is also pseudomonotone.Journal of Convex Analysis, 10(1):149–168, 2003

Monica Bianchi, Nicolas Hadjisavvas, and Siegfried Schaible. On pseudomonotone mapsTfor which−T is also pseudomonotone.Journal of Convex Analysis, 10(1):149–168, 2003

2003

[4] [4]

Asplund decomposition of monotone operators.SIAM Journal on Optimization, 18(3):946–960, 2007

Jonathan Borwein and Herre Wiersma. Asplund decomposition of monotone operators.SIAM Journal on Optimization, 18(3):946–960, 2007

2007

[5] [5]

Nesterov Acceleration with Operator Decomposition

Jaewook Lee, Ernest K. Ryu, and Chulhee Yun. Nesterov acceleration with operator decomposition. arXiv preprint arXiv:2604.11105, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[6] [6]

R. T. Rockafellar.Monotone operators associated with saddle-functions and minimax problems, volume 18.1, page 241–250. American Mathematical Society, 1970

1970

[7] [7]

Tyrrell Rockafellar.Convex analysis, volume 28

R. Tyrrell Rockafellar.Convex analysis, volume 28. Princeton University Press, 1997. 3

1997