[论文] Nonmaximal sums of maximally monotone operators under Rockafellar's constraint qualification (arXiv:2609.10487)

论文概要

研究领域: ML
作者: Weifeng Yang
发布时间: 2026-09-09
arXiv: 2609.10487

中文摘要

本文构造了Rockafellar和猜想反例，其中两个极大单调算子满足内部域条件但它们的和不是极大单调的。作者在c_0和ℓ^1上各给出一个反例，并建立了一个一般构造定理，计算一类图的完整单调极，给出其极大单调性的充要条件，并展示正秩一扰动如何在该条件下产生非极大和。

原文摘要

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 \(\ell^1\).

自动采集于 2026-09-11

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