A new condition for maximal monotonicity via representative functions

Abstract In this paper we give a weaker sufficient condition for the maximal monotonicity of the operator S + A ∗ T A , where S : X ⇉ X ∗ , T : Y ⇉ Y ∗ are two maximal monotone operators, A : X → Y is a linear continuous mapping and X , Y are reflexive Banach spaces. We prove that our condition is weaker than the generalized interior-point conditions given so far in the literature. This condition is formulated using the representative functions of the operators involved. In particular, we rediscover some sufficient conditions given in the past using the so-called Fitzpatrick function for the maximal monotonicity of the sum of two maximal monotone operators and for the precomposition of a maximal monotone operator with a linear operator, respectively.

[1]  Sorin-Mihai Grad,et al.  Maximal Monotonicity for the Precomposition with a Linear Operator , 2006, SIAM J. Optim..

[2]  R. Burachik,et al.  On a Family of Convex Functions associated to Subdifferentials , 2005 .

[3]  Constantin Zălinescu,et al.  A New Proof of the Maximal Monotonicity of the Sum Using the Fitzpatrick Function , 2005 .

[4]  Is Convexity Useful for the Study of Monotonicity , 2003 .

[5]  Sorin-Mihai Grad,et al.  Weaker Constraint Qualifications in Maximal Monotonicity , 2007 .

[6]  R. Rockafellar.  On the maximality of sums of nonlinear monotone operators , 1970 .

[7]  Jean-Paul Penot,et al.  The relevance of convex analysis for the study of monotonicity , 2004 .

[8]  Hristo S. Sendov,et al.  Fitzpatrick functions: inequalities, examples, and remarks on a problem by S. Fitzpatrick , 2005 .

[9]  Teemu Pennanen,et al.  Dualization of Generalized Equations of Maximal Monotone Type , 1999, SIAM J. Optim..

[10]  S. Simons.  Minimax and monotonicity , 1998 .

[11]  S. Fitzpatrick.  Representing monotone operators by convex functions , 1988 .

[12]  R. Rockafellar,et al.  On the maximal monotonicity of subdifferential mappings. , 1970 .

[13]  C. Zălinescu.  Convex analysis in general vector spaces , 2002 .

[14]  R. Phelps.  Lectures on maximal monotone operators , 1993, math/9302209.

[15]  Gert Wanka,et al.  A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces , 2006 .

[16]  C. Zalinescu,et al.  Some problems about the representation of monotone operators by convex functions , 2005 .

[17]  Heinz H. Bauschke.  Fenchel duality, Fitzpatrick functions and the extension of firmly nonexpansive mappings , 2006 .

[18]  H. Attouch,et al.  SOMME PONCTUELLE D’OPÉRATEURS MAXIMAUX MONOTONES , 2009 .

[19]  Jean-Paul Penot.  A representation of maximal monotone operators by closed convex functions and its impact on calculus rules , 2004 .

[20]  F. Browder.  Nonlinear maximal monotone operators in Banach space , 1968 .

[21]  S. K. Mishra,et al.  Nonconvex Optimization and Its Applications , 2008 .

[22]  Stephen Simons,et al.  Fenchel duality, Fitzpatrick functions and the Kirszbraun-Valentine extension theorem , 2005 .

[23]  Benar Fux Svaiter,et al.  Maximal Monotone Operators, Convex Functions and a Special Family of Enlargements , 2002 .

[24]  J. Borwein.  Maximal Monotonicity via Convex Analysis , 2006 .

[25]  Liang Ju Chu.  On the sum of monotone operators , 1996 .