Nonmaximal Sums of Maximally Monotone Operators Under Rockafellar's Constraint Qualification (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 two maximally monotone operators that satisfy the interior-domain condition but whose sum is not maximally monotone. One counterexample is given on the Banach space c_0 and another on l^1 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 yields a nonmaximal sum under this condition. The theorem's hypotheses and maximality criterion are verified on c_0, yielding a counterexample to the conjecture. Additionally, a bounded linear surjection from l^1 onto c_0 is constructed and used to obtain the counterexample on l^1. The work is relevant to monotone operator theory and its applications in optimization and machine learning.

Paper Overview

Research area: ML Author: Weifeng Yang Published: 2026-09-09 arXiv: 2609.10487

Abstract (translated)

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

Explicit counterexamples to Rockafellar's sum conjecture under the interior-domain constraint qualification, on both $$c_0$$ and $\ell^1$ .
A general construction theorem computing the full monotone polar of a class of graphs.
A necessary and sufficient condition for maximal monotonicity of these graphs, and a mechanism whereby positive rank-one perturbations produce nonmaximal sums.
A bounded linear surjection $\ell^1 \to c_0$ used to transfer the counterexample to $\ell^1$ .

Link: https://arxiv.org/abs/2609.10487

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