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

• URL: http://arxiv.org/abs/2609.10487v2
• Date: Sat, 12 Sep 2026 07:11:29 GMT
• Title: Nonmaximal sums of maximally monotone operators under Rockafellar's constraint qualification
• Abstract summary: 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.<n>We compute 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.
• Score: 0.701633634930651
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $\ell^1$. Lean formalizations of the $c_0$ counterexample and the pullback lemma are also provided.

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