Maximally monotone operator sums break under common domain condition
Nonmaximal sums of maximally monotone operators under Rockafellar's constraint qualification
Machine Learning
Summary
Usually, when two mathematical rules called maximally monotone operators meet certain friendly conditions, their combined rule keeps those properties. This paper shows surprising examples where even if these conditions are met, the combined rule does not behave as expected. The authors build these examples in special mathematical spaces and develop tools to understand why the usual guarantee fails. Their findings highlight limits in a well-known assumption about combining these operators.
maximally monotone operatorRockafellar's constraint qualificationsum conjectureinterior-domain conditionmonotone polarpositive rank-one perturbationBanach spacec0 spacel1 space
Authors
Weifeng Yang
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 $\ell^1$.