Generalized monotone operators and polarity approach to generalized monotone sets

Author
Mohammad Alizadeh
IASBS
Abstract
Introduction

Many Suppose that is a Banach Space with topological dual space We will denote by the duality pairing between X and . For , we denote by the boundary points of Ω and by the interior of Ω. Also we will denote by the real nonnegative numbers. Let be a set-valued map from to . The domain and graph of are, respectively, defined by



We recall that a set valued operator is monotone if for all and For two multivalued operators and we write ifis an extension of , i.e., . A monotone operator is called maximal monotone if it has no monotone extension other than itself.

In 1988, The Fitzpatrick function of a monotone operator was introduced by Fitzpatrick. The Fitzpatrick function makes a bridge between the results of convex functions and results on maximal monotone operators. For a monotone operator , its Fitzpatrick function is defined by



It is a convex and norm to weak lower semicontinuous and function.

Let be an extended real-valued function. Its effective domain is defined by The function is called proper if . Let be a proper function. The subdifferential (in the sense of Convex Analysis) of at is defined by



Given a proper function and a map , then is called -convex if



For all and for all

Given an operator and a map . Then is called -monotone if for all and we have



Also is called maximal -monotone if it has no -monotone extension other than itself. We recall for a proper function the -subdifferential of at is defined by



and if .

Main results

The definition we use for the Fitzpatrick function is the same as for monotone operators.

Assume that is a proper -convex function its conjugate is defined by



First we have the following refinement of the Fenchel-Moreau inequality:



where is the indicator function and .

Also we have the following refinement, when is a proper, -convex and lower semicontinuous function and is a maximal -monotone operator:





Moreover, we approach generalized monotonicity from the point of view of the classical concept of polarity. Besides, we introduce and study the notion of generalized monotone polar of a set A. Moreover, we find some equivalent relations between polarity and maximal generalized monotonicity.
Keywords
Monotone and sigma-monotone operators
Fitzpatrik function
Fenchel inequality
Monotone polar
1. M. H. Alizadeh, Fitzpatrick function for generalized monotone operators, J. Nonlinear Convex Anal. (to appear).

2. M. H. Alizadeh, On generalized convex functions and generalized subdifferential, Optim.Lett. 14 (2020) 157-169.

3. M. H. Alizadeh, M .Roohi, Some results on pre-monotone operators, Bull. Iran. Math. Soc. 43 (2017) 2085-2097.

4. M. H. Alizadeh, N. Hadjisavvas, M. Roohi, Local boundedness properties for generalized monotone operators, J. Convex Anal. 19 (2012) 49-61.

5. H. H. Bauschke, D. A. McLaren, H. S. Sendov, Fitzpatrick functions: inequalities, examples, and remarks on a problem by S. Fitzpatrick, J. Convex Anal. 13, (2006) 499-523.

6. J. M. Borwein, Fifty years of maximal monotonicity. Optim. Lett. 4 (2010) 473-490.

7. O. Bueno, J. Cotrina, The pseudomonotone polar for multivalued operators, Optimization 66 (2017) 691-703.

8. O. Bueno, J.-E. Martinez-Legaz, B. F. Svaiter, On the monotone polar and representable closures of monotone operators, J. Convex Anal. 21 (2014)495-505.

9. P. Cheraghi, A.P. Farajzadeh, S. Suantai, On optimization via ϵ-generalized weak subdifferentials, Thai Journal of Mathematics 16(1) 147-164, (2018).

10. A. Farajzadeh, P. Cheraghi , On optimality conditions via weak subdifferential and augmented normal cone, Iranian Journal of Operations Research 9 (2), 15-30, 2018.

11. S. Fitzpatrick, Representing monotone operators by convex functions, in: FunctionalAnalysis and Optimization, Workshop/Miniconference (Canberra1988), Proc. Cent. Math. Anal. Aust. Natl. Univ. 20, Australian National University, Canberra (1988) 59-65.

12. A. N. Iusem, G. Kassay, W. Sosa, An existence result for equilibrium problems with some surjectivity consequences, J. Convex Anal. 16 (2009) 807-826.

13. A. Jofre, D. T. Luc, M. Thera, -subdifferential and -monotonicity, Nonlinear Anal. 33 (1998) 71-90.

14. D.T.Luc, H. V. Ngai, M. Thera, Approximate convex functions, J. Nonlinear Convex Anal. 1 (2000), 155-176.

15. J.-E. Martinez-Legaz, B. F. Svaiter, Monotone operators representable by l.s.c. convex functions, Set-Valued Anal. 13 (2005)21-46.
Mathematical Research
Volume 8, Issue 4 - Serial Number 23
Autumn 2022
Pages 151-163