Nonmaximal Sums of Maximally Monotone Operators: Counterexamples to Rockafellar's Sum Conjecture (arXiv:2609.10487)

Forum topic · 小凯 · 2026-09-11

Summary

This paper by Weifeng Yang (arXiv:2609.10487) constructs counterexamples to Rockafellar's sum conjecture, showing that the sum of two maximally monotone operators can fail to be maximally monotone even when they satisfy the interior-domain condition. One counterexample is given on the Banach space c0 and another on ℓ1 with its usual norm. The author establishes a general construction theorem that computes the entire monotone polar of a class of graphs, derives a necessary and sufficient condition for their maximal monotonicity, and shows how a positive rank-one perturbation produces a nonmaximal sum under this condition. The theorem's hypotheses and maximality criterion are verified on c0, yielding the counterexample, and a bounded linear surjection from ℓ1 onto c0 is constructed to transfer the result to ℓ1. This work resolves a long-standing open question in monotone operator theory with implications for convex analysis and optimization.

Paper Overview

Field: Mathematical analysis / ML theory Author: Weifeng Yang Published: 2026-09-09 arXiv: 2609.10487

Abstract

We construct counterexamples to Rockafellar's sum conjecture in which two maximally monotone operators satisfy the interior-domain condition but their sum is not maximally monotone. We give one counterexample on $$c_0$$ and another on $\ell^1$ with its usual norm. We establish a general construction theorem that computes the entire monotone polar of a class of graphs, gives a necessary and sufficient condition for their maximal monotonicity, and shows how a positive rank-one perturbation yields a nonmaximal sum under this condition. We verify the theorem's hypotheses and its maximality criterion on $$c_0$$ , thereby obtaining a counterexample to the conjecture. Furthermore, we construct a bounded linear surjection from $\ell^1$ onto $$c_0$$ and use it to obtain the counterexample on $\ell^1$ .

Key Contributions

Counterexamples on $$c_0$$ and $\ell^1$ : Two explicit counterexamples to Rockafellar's sum conjecture, where the interior-domain condition holds but the sum of maximally monotone operators is not maximally monotone.
General construction theorem: Computes the entire monotone polar of a class of graphs and characterizes maximal monotonicity via a necessary and sufficient condition.
Rank-one perturbation mechanism: Shows how a positive rank-one perturbation yields a nonmaximal sum under the stated condition.
Transfer to $\ell^1$ : A bounded linear surjection from $\ell^1$ onto $$c_0$$ is constructed and used to obtain the $\ell^1$ counterexample.

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