Levitin–Polyak well-posedness of variational inequality problems with functional constraints

In this paper, we introduce several types of (generalized) Levitin–Polyak well-posednesses for a variational inequality problem with abstract and functional constraints. Criteria and characterizations for these types of well-posednesses are given. Relations among these types of well-posednesses are also investigated.

[1]  A. N. Tikhonov,et al.  On the stability of the functional optimization problem , 1966 .

[2]  Massimo Furi,et al.  About well-posed optimization problems for functionals in metric spaces , 1970 .

[3]  A. Bensoussan.  Points de Nash Dans le Cas de Fonctionnelles Quadratiques et Jeux Differentiels lineaires a N Personnes , 1974 .

[4]  A. Auslender.  Optimisation : méthodes numériques , 1976 .

[5]  Clermont Dupuis,et al.  An Efficient Method for Computing Traffic Equilibria in Networks with Asymmetric Transportation Costs , 1984, Transp. Sci..

[6]  Ewa M. Bednarczuk.  Well Posedness of Vector Optimization Problems , 1987 .

[7]  Roberto Lucchetti,et al.  Well Posedness, Towards Vector Optimization , 1987 .

[8]  Patrick T. Harker,et al.  Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications , 1990, Math. Program..

[9]  Jean-Paul Penot,et al.  Metrically well-set minimization problems , 1992 .

[10]  Masao Fukushima,et al.  Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems , 1992, Math. Program..

[11]  Roberto Lucchetti,et al.  The EPI-Distance Topology: Continuity and Stability Results with Applications to Convex Optimization Problems , 1992, Math. Oper. Res..

[12]  T. Zolezzi,et al.  Well-Posed Optimization Problems , 1993 .

[13]  J. P. Revalski,et al.  Constrained convex optimization problems-well-posedness and stability * , 1994 .

[14]  T. Zolezzi,et al.  Well-posedness criteria in optimization with application to the calculus of variations , 1995 .

[15]  P. Loridan.  Well-posedness in Vector Optimization , 1995 .

[16]  Fioravante Patrone,et al.  Well-Posedness for Nash Equilibria and Related Topics , 1995 .

[17]  T. Zolezzi.  Extended well-posedness of optimization problems , 1996 .

[18]  Julian P. Revalski.  Hadamard and Strong Well-Posedness for Convex Programs , 1997, SIAM J. Optim..

[19]  Fioravante Patrone,et al.  A New Approach To Tikhonov Well-Posedness For Nash Equilibria ∗ , 1997 .

[20]  Jacqueline Morgan,et al.  Generalized Variational Inequalities with Pseudomonotone Operators Under Perturbations , 1999 .

[21]  A. Auslender,et al.  Asymptotic Analysis for Penalty and Barrier Methods in Variational Inequalities , 1999 .

[22]  X. X. Huang.  Extended Well-Posedness Properties of Vector Optimization Problems , 2000 .

[23]  Tullio Zolezzi.  Well-Posedness and Optimization under Perturbations , 2001, Ann. Oper. Res..

[24]  X. X. Huang,et al.  Extended and strongly extended well-posedness of set-valued optimization problems , 2001, Math. Methods Oper. Res..

[25]  Jacqueline Morgan,et al.  Approximate Solutions and α-Well-Posedness for Variational Inequalities and Nash Equilibria , 2002 .

[26]  Georges Zaccour,et al.  Decision and control in management science , 2002 .

[27]  Daoli Zhu.  Augmented Lagrangian Theory, Duality and Decomposition Methods for Variational Inequality Problems , 2003 .

[28]  Sien Deng.  Coercivity properties and well-posedness in vector optimization , 2003, RAIRO Oper. Res..

[29]  Alfred Auslender,et al.  Variational inequalities over the cone of semidefinite positive symmetric matrices and over the Lorentz cone , 2003, Optim. Methods Softw..

[30]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[31]  Bingsheng He,et al.  A modified augmented Lagrangian method for a class of monotone variational inequalities , 2004, Eur. J. Oper. Res..

[32]  Jian Yu,et al.  Unified Approaches to Well-Posedness with Some Applications , 2005, J. Glob. Optim..

[33]  Xiaoqi Yang,et al.  Vector Optimization: Set-Valued and Variational Analysis , 2005 .

[34]  X. Q. Yang,et al.  Generalized Levitin--Polyak Well-Posedness in Constrained Optimization , 2006, SIAM J. Optim..

[35]  M. Beatrice Lignola,et al.  α-Well-posedness for Nash Equilibria and For Optimization Problems with Nash Equilibrium Constraints , 2006, J. Glob. Optim..

[36]  M. B. Lignola.  Well-Posedness and L-Well-Posedness for Quasivariational Inequalities , 2006 .

[37]  Kok Lay Teo,et al.  Calmness and Exact Penalization in Vector Optimization with Cone Constraints , 2006, Comput. Optim. Appl..

[38]  Kok Lay Teo,et al.  Levitin-Polyak Well-Posedness for Equilibrium Problems with Functional Constraints , 2007 .

[39]  M. Fukushima.  A class of gap functions for quasi-variational inequality problems , 2007 .

[40]  X. Q. Yang,et al.  Levitin–Polyak well-posedness of constrained vector optimization problems , 2007, J. Glob. Optim..

[41]  Jen-Chih Yao,et al.  EXISTENCE THEOREMS FOR GENERALIZED VECTOR VARIATIONAL INEQUALITIES WITH PSEUDOMONOTONICITY AND THEIR APPLICATIONS , 2008 .

[42]  Jen-Chih Yao,et al.  Well-Posedness for Mixed Quasivariational-Like Inequalities , 2008 .

[43]  Jen-Chih Yao,et al.  Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems , 2008, J. Glob. Optim..

[44]  X. X. Huang,et al.  Levitin–Polyak well-posedness of generalized quasivariational inequalities with functional constraints , 2009 .

[45]  Shengjie Li,et al.  Levitin–Polyak well-posedness of vector equilibrium problems , 2009, Math. Methods Oper. Res..

[46]  Jen-Chih Yao,et al.  WELL-POSEDNESS FOR VECTOR QUASIEQUILIBRIA , 2009 .

[47]  Jen-Chih Yao,et al.  Well-posedness by perturbations of mixed variational inequalities in Banach spaces , 2010, Eur. J. Oper. Res..