On Nonconvex Pseudomonotone Equilibrium Problems with Applications

In this paper, we provide a further study for nonconvex pseudomonotone equilibrium problems and nonconvex mixed variational inequalities by using global directional derivatives. We provide finer necessary and sufficient optimality conditions for both problems in the pseudomonotone case and, as a consequence, a characterization for a point to be a solution for nonconvex equilibrium problems is given. Finally, we apply the golden ratio algorithm for a class of nonconvex functions in equilibrium problems and mixed variational inequalities.

[1]  J. Penot.  Are Generalized Derivatives Sseful for Generalized Convex Functions , 1998 .

[2]  Alfredo N. Iusem,et al.  Optimality Conditions for Vector Equilibrium Problems with Applications , 2018, J. Optim. Theory Appl..

[3]  A. Cambini,et al.  Generalized Convexity and Optimization , 2009 .

[4]  Michel Théra,et al.  Derivatives with Support and Applications , 1994, Math. Oper. Res..

[5]  John Cotrina,et al.  Equilibrium Problems: Existence Results and Applications , 2018 .

[6]  Felipe Lara,et al.  Optimality Conditions for Nonconvex Nonsmooth Optimization via Global Derivatives , 2020, J. Optim. Theory Appl..

[7]  Khalid Addi,et al.  Complementarity and Variational Inequalities in Electronics , 2017 .

[8]  Vaithilingam Jeyakumar,et al.  A Solvability Theorem for a Class of Quasiconvex Mappings with Applications to Optimization , 1993 .

[9]  I. V. Konnov,et al.  Mixed variational inequalities and economic equilibrium problems , 2002 .

[10]  M. Gowda.  Pseudomonotone and copositive star matrices , 1989 .

[11]  Jean-Pierre Crouzeix,et al.  Pseudomonotone variational inequality problems: Existence of solutions , 1997, Math. Program..

[12]  G. Giorgi,et al.  Mathematics of Optimization: Smooth and Nonsmooth Case , 2004 .

[13]  Alfredo N. Iusem,et al.  A Note on "Existence Results for Noncoercive Mixed Variational Inequalities in Finite Dimensional Spaces" , 2020, J. Optim. Theory Appl..

[14]  N. T. Vinh.  Golden ratio algorithms for solving equilibrium problems in Hilbert spaces , 2018, 1804.01829.

[15]  W. Oettli.  A remark on vector-valued equilibria and generalized monotonicity , 1997 .

[16]  Alfredo N. Iusem,et al.  On certain conditions for the existence of solutions of equilibrium problems , 2008, Math. Program..

[17]  M. Teboulle,et al.  Asymptotic cones and functions in optimization and variational inequalities , 2002 .

[18]  C. S. Lalitha,et al.  Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization , 2013 .

[19]  Siegfried Schaible,et al.  Handbook of Generalized Convexity and Generalized Monotonicity , 2005 .

[20]  F. Flores-Bazán.  Existence Theorems for Generalized Noncoercive Equilibrium Problems: The Quasi-Convex Case , 2000 .

[21]  Andrew Craig Eberhard,et al.  Generalized Convexity, Generalized Monotonicity and Applications , 2005 .

[22]  Yura Malitsky,et al.  Golden ratio algorithms for variational inequalities , 2018, Mathematical Programming.

[23]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[24]  Min Wang.  The existence results and Tikhonov regularization method for generalized mixed variational inequalities in Banach spaces , 2017 .

[25]  R. Rockafellar.  Generalized Directional Derivatives and Subgradients of Nonconvex Functions , 1980, Canadian Journal of Mathematics.

[26]  Fabián Flores Bazán,et al.  Maximizing and minimizing quasiconvex functions: related properties, existence and optimality conditions via radial epiderivatives , 2015, J. Glob. Optim..

[27]  Felipe Lara,et al.  A Quasiconvex Asymptotic Function with Applications in Optimization , 2018, J. Optim. Theory Appl..

[28]  Jean-Baptiste Hiriart-Urruty,et al.  What is the subdifferential of the closed convex hull of a function , 1996 .