Optimality and Duality of Approximate Quasi Weakly Efficient Solution for Nonsmooth Vector Optimization Problems

This paper aims at studying optimality conditions and duality theorems of an approximate quasi weakly efficient solution for a class of nonsmooth vector optimization problems (VOP). First, a necessary optimality condition to the problem (VOP) is established by using the Clarke subdifferential. Second, the concept of approximate pseudo quasi type-I function is introduced, and under its hypothesis, a sufficient optimality condition to the problem (VOP) is also obtained. Finally, the approximate Mond–Weir dual model of the problem (VOP) is presented, and then, weak, strong, and converse duality theorems are established.

[1]  San-yang Liu,et al.  Generalized Lagrangian Duality in Set-valued Vector Optimization via Abstract Subdifferential , 2022, Acta Mathematicae Applicatae Sinica, English Series.

[2]  Sunila Sharma,et al.  Optimality and duality in nonsmooth vector optimization with non-convex feasible set , 2020, RAIRO Oper. Res..

[3]  Jing Zeng,et al.  Characterizations of Approximate Duality and Saddle Point Theorems for Nonsmooth Robust Vector Optimization , 2020, Numerical Functional Analysis and Optimization.

[4]  Ram N. Mohapatra,et al.  On the Applications of Nonsmooth Vector Optimization Problems to Solve Generalized Vector Variational Inequalities Using Convexificators , 2019, WCGO.

[5]  T. D. Chuong,et al.  Subdifferentials and derivatives with respect to a set and applications to optimization , 2019 .

[6]  Jen-Chih Yao,et al.  Optimality Conditions and Duality for Robust Nonsmooth Multiobjective Optimization Problems with Constraints , 2018, J. Optim. Theory Appl..

[7]  C. Wen,et al.  Optimality Conditions for Approximate Pareto Solutions of a Nonsmooth Vector Optimization Problem with an Infinite Number of Constraints , 2018, Acta Mathematica Vietnamica.

[8]  Shashi Kant Mishra,et al.  On minty variational principle for nonsmooth vector optimization problems with approximate convexity , 2016, Optim. Lett..

[9]  Shashi Kant Mishra,et al.  On approximate starshapedness in multiobjective optimization , 2016, Optim. Methods Softw..

[10]  Chandal Nahak,et al.  Optimality conditions for approximate quasi efficiency in set-valued equilibrium problems , 2016 .

[11]  Nguyen Minh Tung,et al.  Optimality conditions and duality for nonsmooth vector equilibrium problems with constraints , 2015 .

[12]  Yanyan Feng,et al.  Optimality conditions for vector equilibrium problems with constraint in Banach spaces , 2014, Optim. Lett..

[13]  Shashi Kant Mishra,et al.  On Approximately Star-Shaped Functions and Approximate Vector Variational Inequalities , 2013, J. Optim. Theory Appl..

[14]  Shashi Kant Mishra,et al.  Some relations between vector variational inequality problems and nonsmooth vector optimization problems using quasi efficiency , 2013 .

[15]  T. Antczak.  Proper efficiency conditions and duality results for nonsmooth vector optimization in Banach spaces under (Φ,ρ)-invexity , 2012 .

[16]  Xinmin Yang.  On symmetric and self duality in vector optimization problem , 2011 .

[17]  Bienvenido Jiménez,et al.  Strong Kuhn–Tucker conditions and constraint qualifications in locally Lipschitz multiobjective optimization problems , 2009 .

[18]  Majid Soleimani-Damaneh,et al.  Nonsmooth Optimization Using Mordukhovich's Subdifferential , 2009, SIAM J. Control. Optim..

[19]  S. J. Li,et al.  Different Conjugate Dual Problems in Vector Optimization and Their Relations , 2009 .

[20]  A. Mehra,et al.  Approximate convexity in vector optimisation , 2006, Bulletin of the Australian Mathematical Society.

[21]  X. Li,et al.  Stronger Kuhn-Tucker Type Conditions in Nonsmooth Multiobjective Optimization: Locally Lipschitz Case , 2005 .

[22]  A. Taa,et al.  Subdifferentials of multifunctions and Lagrange multipliers for multiobjective optimization , 2003 .

[23]  L. Thibault,et al.  The approximate subdifferential of composite functions , 1993, Bulletin of the Australian Mathematical Society.

[24]  C. Gerth,et al.  Nonconvex separation theorems and some applications in vector optimization , 1990 .

[25]  Jay S. Treiman,et al.  Generalized Gradients, Lipschitz behavior and Directional Derivatives , 1985, Canadian Journal of Mathematics.

[26]  R. Rockafellar.  Augmented Lagrange Multiplier Functions and Duality in Nonconvex Programming , 1974 .

[27]  D. Bhatia,et al.  Optimality via generalized approximate convexity and quasiefficiency , 2013, Optim. Lett..