2009/02/06 by Heinz H. Bauschke, Bauschke, Heinz H., Xianfu Wang +3
Computer Science · Mathematics · #47A06 #47H05 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Inequalities and Applications #Optimization and Control (math.OC) #Optimization and Variational Analysis #Point processes and geometric inequalities #math.FA #math.OC #msc:47A06 #msc:47H05
paper · pdf · doi:10.48550/arxiv.0902.1189
arxiv created 2009/02/06 · openalex publication_date 2009/02/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The question whether or not the sum of two maximal monotone operators is maximal monotone under Rockafellar's constraint qualification - that is, whether or not "the sum theorem" is true - is the most famous open problem in Monotone Operator Theory. In his 2008 monograph "From Hahn-Banach to Monotonicity", Stephen Simons asked whether or not the sum theorem holds for the special case of a maximal monotone linear operator and a normal cone operator of a closed convex set provided that the interior of the set makes a nonempty intersection with the domain of the linear operator. In this note, we provide an affirmative answer to Simons' question. In fact, we show that the sum theorem is true for a maximal monotone linear relation and a normal cone operator. The proof relies on Rockafellar's formula for the Fenchel conjugate of the sum as well as some results featuring the Fitzpatrick function.