The Sum of a Maximally Monotone Linear Relation and the Subdifferential of a Proper Lower Semicontinuous Convex Function is Maximally Monotone

Abstract

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 + ∂f provided that A is a maximally monotone linear relation, and f is a proper lower semicontinuous convex function satisfying dom A ∩ int dom ∂f = ∅. Moreover, A + ∂f is of type (FPV). The maximal monotonicity of A + ∂f when int dom A ∩ dom ∂f = ∅ follows from a result by Verona and Verona, which the present work complements.2010 Mathematics Subject Classification: Primary 47A06, 47H05; Secondary 47B65, 47N10, 90C25

Recent progress on Monotone Operator Theory

“…A positive answer to various restricted versions of (i) implies a positive answer to (ii) [21,65]. Some recent developments on the sum problem can be found in Simons' monograph [65] and [12,13,15,21,24,80,46,72,81,83]. In [24], we showed if the following conjecture is true then the sum problem would have an affirmative answer.…”

On sum of monotone operator of type (FPV) and a maximal monotone operator

“…In [2], Bauschke, Wang and Yao prove that the sum of maximal monotone linear relation and the subdifferential operator of a sublinear function with Rockafellar's constraint qualification is maximal monotone. In [15], Yao extend the results in [2] to the subdifferential operator of any proper lower semicontinuous convex function. Yao [16] proves the that the sum of two maximal monotone operators A and B satisfying the conditions A + N domB is of type (FPV) and domA ∩ int domB = φ is maximal.…”

Monotone Operators Without Enlargements

“…(45) Then by (40) and (45), z = y * * . Thus by (45), F A+N C (z, z * ) ≥ g * (y * , z) + ι * C (z * − y * ) = F A (z, y * ) + ι * C (z * − y * ) = F A (z, y * ) + ι * C (z * − y * ) + ι C (z) (by (41)) = F A (z, y * ) + F N C (z, z * − y * ) (by Fact 3.19).…”