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SUM THEOREMS FOR MAXIMALLY MONOTONE OPERATORS OF TYPE (FPV)

Part of: General convexity Miscellaneous topics in calculus of variations and optimal control Nonlinear operators and their properties

Published online by Cambridge University Press:  08 July 2014

JONATHAN M. BORWEIN  and
LIANGJIN YAO
JONATHAN M. BORWEIN
Affiliation:
CARMA, University of Newcastle, Newcastle, New South Wales 2308, Australia email jonathan.borwein@newcastle.edu.au
LIANGJIN YAO*
Affiliation:
CARMA, University of Newcastle, Newcastle, New South Wales 2308, Australia email liangjin.yao@newcastle.edu.au
*
✉liangjin.yao@newcastle.edu.au
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Abstract

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The most important open problem in monotone operator theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that the classical Rockafellar’s constraint qualification holds. In this paper, we establish the maximal monotonicity of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}A+B$ provided that $A$ and $B$ are maximally monotone operators such that ${\rm star}({\rm dom}\ A)\cap {\rm int}\, {\rm dom}\, B\neq \varnothing $, and $A$ is of type (FPV). We show that when also ${\rm dom}\ A$ is convex, the sum operator $A+B$ is also of type (FPV). Our result generalizes and unifies several recent sum theorems.

Keywords

constraint qualificationconvex functionconvex setFitzpatrick functionlinear relationmaximally monotone operatormonotone operatoroperator of type (FPV)sum problem

MSC classification

Secondary: 47H05: Monotone operators and generalizations 49N15: Duality theory 52A41: Convex functions and convex programs 90C25: Convex programming
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 97 , Issue 1 , August 2014 , pp. 1 - 26
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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