scholarly journals Higher-Order Weakly Generalized Epiderivatives and Applications to Optimality Conditions

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Qilin Wang ◽  
Guolin Yu
Keyword(s):  
Optimality Conditions ◽  
Optimization Problem ◽  
Efficient Solutions ◽  
Higher Order ◽  
Sufficient Optimality Conditions ◽  
Contingent Epiderivative ◽  
Constraint Set ◽  
Necessary And Sufficient ◽  
Henig Efficient Solutions ◽  
Generalized Contingent Epiderivative

The notions of higher-order weakly generalized contingent epiderivative and higher-order weakly generalized adjacent epiderivative for set-valued maps are proposed. By virtue of the higher-order weakly generalized contingent (adjacent) epiderivatives, both necessary and sufficient optimality conditions are obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map. The imposed assumptions are relaxed in comparison with those of recent results in the literature. Examples are provided to show some advantages of our notions and results.

scholarly journals Optimality conditions for robust weak sharp efficient solutions of nonsmooth uncertain multiobjective optimization problems

2021 ◽  
Author(s):  
Jutamas Kerdkaew ◽  
Rabian Wangkeeree ◽  
Rattanaporn Wangkeereee
Keyword(s):  
Multiobjective Optimization ◽  
Optimality Conditions ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Efficient Solutions ◽  
Sufficient Optimality Conditions ◽  
Nonconvex Functions ◽  
Multiobjective Optimization Problems ◽  
Necessary And Sufficient ◽  
Weak Sharp

AbstractIn this paper, we investigate an uncertain multiobjective optimization problem involving nonsmooth and nonconvex functions. The notion of a (local/global) robust weak sharp efficient solution is introduced. Then, we establish necessary and sufficient optimality conditions for local and/or the robust weak sharp efficient solutions of the considered problem. These optimality conditions are presented in terms of multipliers and Mordukhovich/limiting subdifferentials of the related functions.

OPTIMALITY OF GLOBAL PROPER EFFICIENCY FOR CONE-ARCWISE CONNECTED SET-VALUED OPTIMIZATION USING CONTINGENT EPIDERIVATIVE

2013 ◽  
Vol 30 (03) ◽  
pp. 1340004 ◽  
Author(s):  
GUOLIN YU
Keyword(s):  
Optimality Conditions ◽  
Optimization Problem ◽  
Vector Optimization Problem ◽  
Proper Efficiency ◽  
Sufficient Optimality Conditions ◽  
Set Valued Mapping ◽  
Contingent Epiderivative ◽  
Connected Set ◽  
Arcwise Connected ◽  
Necessary And Sufficient

This note deals with the optimality conditions of set-valued unconstraint optimization problem in real normed linear spaces. Based upon the concept of contingent epiderivative, the unified necessary and sufficient optimality conditions for global proper efficiency in vector optimization problem involving cone-arcwise connected set-valued mapping are presented.

scholarly journals Robust Approximate Optimality Conditions for Uncertain Nonsmooth Optimization with Infinite Number of Constraints

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 12 ◽  
Author(s):  
Xiangkai Sun ◽  
Hongyong Fu ◽  
Jing Zeng
Keyword(s):  
Optimality Conditions ◽  
Optimization Problem ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Optimization Technique ◽  
Optimal Solution ◽  
Sufficient Optimality Conditions ◽  
Convex Constraint ◽  
Approximate Optimality Conditions ◽  
Necessary And Sufficient

This paper deals with robust quasi approximate optimal solutions for a nonsmooth semi-infinite optimization problems with uncertainty data. By virtue of the epigraphs of the conjugates of the constraint functions, we first introduce a robust type closed convex constraint qualification. Then, by using the robust type closed convex constraint qualification and robust optimization technique, we obtain some necessary and sufficient optimality conditions for robust quasi approximate optimal solution and exact optimal solution of this nonsmooth uncertain semi-infinite optimization problem. Moreover, the obtained results in this paper are applied to a nonsmooth uncertain optimization problem with cone constraints.

scholarly journals Higher-order optimality conditions for a minimax

1996 ◽  
Vol 54 (3) ◽  
pp. 509-516 ◽  
Author(s):  
Do Van Luu ◽  
W. Oettli
Keyword(s):  
Optimality Conditions ◽  
Minimax Problem ◽  
Higher Order ◽  
Regularity Conditions ◽  
Sufficient Optimality Conditions ◽  
Basic Assumptions ◽  
Necessary And Sufficient ◽  
Inequality Type

Higher-order necessary and sufficient optimality conditions for a nonsmooth minimax problem with infinitely many constraints of inequality type are established under suitable basic assumptions and regularity conditions.

scholarly journals Optimality and duality in set-valued optimization utilizing limit sets

2018 ◽  
Vol 16 (1) ◽  
pp. 1128-1139
Author(s):  
Xiangyu Kong ◽  
Yinfeng Zhang ◽  
GuoLin Yu
Keyword(s):  
Optimality Conditions ◽  
Duality Theory ◽  
Optimization Problem ◽  
Convex Set ◽  
Sufficient Optimality Conditions ◽  
Constraint Optimization ◽  
Limit Sets ◽  
Optimality And Duality ◽  
Wolfe Type Dual ◽  
Necessary And Sufficient

AbstractThis paper deals with optimality conditions and duality theory for vector optimization involving non-convex set-valued maps. Firstly, under the assumption of nearly cone-subconvexlike property for set-valued maps, the necessary and sufficient optimality conditions in terms of limit sets are derived for local weak minimizers of a set-valued constraint optimization problem. Then, applications to Mond-Weir type and Wolfe type dual problems are presented.

scholarly journals Higher order optimality and duality in fractional vector optimization over cones

2017 ◽  
Vol 48 (3) ◽  
pp. 273-287 ◽  
Author(s):  
Muskan Kapoor ◽  
Surjeet Kaur Suneja ◽  
Meetu Bhatia Grover
Keyword(s):  
Optimality Conditions ◽  
Vector Optimization ◽  
Convex Functions ◽  
Optimization Problem ◽  
Vector Optimization Problem ◽  
Higher Order ◽  
Sufficient Optimality Conditions ◽  
Dual Program ◽  
Duality Results ◽  
Optimality And Duality

In this paper we give higher order sufficient optimality conditions for a fractional vector optimization problem over cones, using higher order cone-convex functions. A higher order Schaible type dual program is formulated over cones.Weak, strong and converse duality results are established by using the higher order cone convex and other related functions.

scholarly journals Vector optimization with cone semilocally preinvex functions

2013 ◽  
Vol 4 (1) ◽  
pp. 11-20 ◽  
Author(s):  
Surjeet Kaur Suneja ◽  
Meetu Bhatia
Keyword(s):  
Optimality Conditions ◽  
Vector Optimization ◽  
Minimization Problem ◽  
Optimization Problem ◽  
Vector Optimization Problem ◽  
Sufficient Optimality Conditions ◽  
Duality Theorems ◽  
Preinvex Functions ◽  
Necessary And Sufficient

In this paper we introduce cone semilocally preinvex, cone semilocally quasi preinvex and cone semilocally pseudo preinvex functions and study their properties. These functions are further used to establish necessary and sufficient optimality conditions for a vector minimization problem over cones. A Mond-Weir type dual is formulated for the vector optimization problem and various duality theorems are proved.

Necessary and sufficient optimality conditions for set-valued optimization problems

2011 ◽  
Vol 18 (1) ◽  
pp. 53-66
Author(s):  
Najia Benkenza ◽  
Nazih Gadhi ◽  
Lahoussine Lafhim
Keyword(s):  
Optimality Conditions ◽  
Vector Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Necessary Optimality Conditions ◽  
Sufficient Optimality Conditions ◽  
Necessary And Sufficient ◽  
First Time ◽  
Funct Anal

Abstract Using a special scalarization employed for the first time for the study of necessary optimality conditions in vector optimization by Ciligot-Travain [Numer. Funct. Anal. Optim. 15: 689–693, 1994], we give necessary optimality conditions for a set-valued optimization problem by establishing the existence of Lagrange–Fritz–John multipliers. Also, sufficient optimality conditions are given without any Lipschitz assumption.

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