Counterexamples show failure of sum rule for monotone operators
Nonmaximal sums of maximally monotone operators under Rockafellar's constraint qualification
Machine Learning
Summary
Sometimes in math, when you combine two objects that each have a special 'maximally monotone' property, the combination should also have that property. This was a long-standing idea thought to always be true under certain conditions, called Rockafellar's sum conjecture. The authors found explicit cases where the combination loses this property, contradicting the old belief. They proved this by creating specific examples in certain function spaces and developed a method to understand when these properties hold or fail.
What this means in practice
- •For optimization engineers: Recognize limitations of combining maximal monotone operators when designing algorithms relying on operator sums under interior-domain conditions.
- •For functional analysts: Use the explicit counterexamples and criteria to better understand monotonicity properties in infinite-dimensional spaces to refine theoretical models.
A theory result. No direct application yet.
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 $\ell^1$.