The sum of a maximally monotone linear relation and the subdifferential of a proper lower semicontinuous convex function is maximally monotone

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar’s constraint qualification holds. In this paper, we prove the maximal monotonicity of $A+\mathrm{\partial }f$ provided that $A$ is a maximally monotone linear relation, and $f$ is a proper lower semicontinuous convex function satisfying $\dom A\cap\inte\dom \partial f\neq\varnothing$\mathrm{.}Moreover,$A+\partial f$ is of type (FPV). The maximal monotonicity of $A+\mathrm{\partial }f$ when $\intdom A\cap\dom \partial f\neq\varnothing$ follows from a result by Verona and Verona, which the present work complements.