Nonmaximal sums of maximally monotone operators under Rockafellar's constraint qualification

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 $\ell^1$.

Machine Learning (cs.LG); Functional Analysis (math.FA)

arXiv:2609.10487 [cs.LG]

(or arXiv:2609.10487v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.10487 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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