scholarly journals Multiobjective programming under nondifferentiable G-V-invexity

Filomat ◽  
2016 ◽  
Vol 30 (11) ◽  
pp. 2909-2923 ◽  
Author(s):  
Tadeusz Antczak
Keyword(s):  
Optimality Conditions ◽  
Programming Problem ◽  
Dual Problem ◽  
Multiobjective Programming ◽  
Necessary Optimality Conditions ◽  
Vector Optimization Problem ◽  
Alternative Theorem ◽  
Nonsmooth Multiobjective Programming ◽  
Locally Lipschitz ◽  
Multiobjective Programming Problem

In the paper, new Fritz John type necessary optimality conditions and new Karush-Kuhn-Tucker type necessary opimality conditions are established for the considered nondifferentiable multiobjective programming problem involving locally Lipschitz functions. Proofs of them avoid the alternative theorem usually applied in such a case. The sufficiency of the introduced Karush-Kuhn-Tucker type necessary optimality conditions are proved under assumptions that the functions constituting the considered nondifferentiable multiobjective programming problem are G-V-invex with respect to the same function ?. Further, the so-called nondifferentiable vector G-Mond-Weir dual problem is defined for the considered nonsmooth multiobjective programming problem. Under nondifferentiable G-V-invexity hypotheses, several duality results are established between the primal vector optimization problem and its G-dual problem in the sense of Mond-Weir.

scholarly journals Duality for a class of nonsmooth multiobjective programming problems using convexificators

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 489-498 ◽  
Author(s):  
Anurag Jayswal ◽  
Krishna Kummari ◽  
Vivek Singh
Keyword(s):  
Optimality Conditions ◽  
Programming Problem ◽  
Multiobjective Programming ◽  
Optimization Problems ◽  
Duality Results ◽  
Nonsmooth Multiobjective Programming ◽  
Multiobjective Programming Problem ◽  
Multiobjective Programming Problems ◽  
Dual Models

As duality is an important and interesting feature of optimization problems, in this paper, we continue the effort of Long and Huang [X. J. Long, N. J. Huang, Optimality conditions for efficiency on nonsmooth multiobjective programming problems, Taiwanese J. Math., 18 (2014) 687-699] to discuss duality results of two types of dual models for a nonsmooth multiobjective programming problem using convexificators.

scholarly journals Sufficiency and Duality in Nonsmooth Multiobjective Programming Problem under Generalized Univex Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Pallavi Kharbanda ◽  
Divya Agarwal ◽  
Deepa Sinha
Keyword(s):  
Optimality Conditions ◽  
Programming Problem ◽  
Multiobjective Programming ◽  
Generalized Functions ◽  
Sufficient Optimality Conditions ◽  
Type I ◽  
Duality Theorems ◽  
Nonsmooth Multiobjective Programming ◽  
Strong Converse ◽  
Multiobjective Programming Problem

We consider a nonsmooth multiobjective programming problem where the functions involved are nondifferentiable. The class of univex functions is generalized to a far wider class of (φ,α,ρ,σ)-dI-V-type I univex functions. Then, through various nontrivial examples, we illustrate that the class introduced is new and extends several known classes existing in the literature. Based upon these generalized functions, Karush-Kuhn-Tucker type sufficient optimality conditions are established. Further, we derive weak, strong, converse, and strict converse duality theorems for Mond-Weir type multiobjective dual program.

scholarly journals Minimax fractional programming with nondiferentiable (G,β)-invexity

Filomat ◽  
2014 ◽  
Vol 28 (10) ◽  
pp. 2027-2035 ◽  
Author(s):  
Xiaoling Liu ◽  
Dehui Yuan
Keyword(s):  
Optimality Conditions ◽  
Programming Problem ◽  
Fractional Programming ◽  
Dual Problem ◽  
Necessary Optimality Conditions ◽  
Minimax Programming ◽  
Fractional Programming Problem ◽  
Duality Results ◽  
Locally Lipschitz ◽  
Minimax Fractional Programming

In this paper, we consider the minimax fractional programming Problem (FP) in which the functions are locally Lipschitz (G,?)-invex. With the help of a useful auxiliary minimax programming problem, we obtain not only G-sufficient but also G-necessary optimality conditions theorems for the Problem (FP). With G-necessary optimality conditions and (G,?)-invexity in the hand, we further construct dual Problem (D) for the primal one (FP) and prove duality results between Problems (FP) and (D). These results extend several known results to a wider class of programs.

scholarly journals Generalized Minimax Programming with Nondifferentiable (G,β)-Invexity

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
D. H. Yuan ◽  
X. L. Liu
Keyword(s):  
Optimality Conditions ◽  
Programming Problem ◽  
Dual Problem ◽  
Necessary Optimality Conditions ◽  
Minimax Programming ◽  
Duality Results ◽  
Locally Lipschitz

We consider the generalized minimax programming problem (P) in which functions are locally Lipschitz (G,β)-invex. Not onlyG-sufficient but alsoG-necessary optimality conditions are established for problem (P). WithG-necessary optimality conditions and (G,β)-invexity on hand, we construct dual problem (DI) for the primal one (P) and prove duality results between problems (P) and (DI). These results extend several known results to a wider class of programs.

AN η-APPROXIMATION APPROACH IN NONLINEAR VECTOR OPTIMIZATION WITH UNIVEX FUNCTIONS

2006 ◽  
Vol 23 (04) ◽  
pp. 525-542 ◽  
Author(s):  
TADEUSZ ANTCZAK
Keyword(s):  
Programming Problem ◽  
Vector Optimization ◽  
Optimization Problem ◽  
Multiobjective Programming ◽  
Sufficient Conditions ◽  
Vector Optimization Problem ◽  
Approximation Approach ◽  
Nonlinear Vector Optimization ◽  
Multiobjective Programming Problem ◽  
Nonlinear Vector

In this paper, the so-called η-approximation approach is used to obtain the sufficient conditions for a nonlinear multiobjective programming problem with univex functions with respect to the same function η. In this method, an equivalent η-approximated vector optimization problem is constructed by a modification of both the objective and the constraint functions in the original multiobjective programming problem at the given feasible point. Moreover, to find the optimal solutions of the original multiobjective problem, it sufficies to solve its associated η-approximated vector optimization problem. Finally, the description of the η-approximation algorithm for solving a nonlinear multiobjective programming problem involving univex functions is presented.

scholarly journals Optimality Conditions for a Nonsmooth Uncertain Multiobjective Programming Problem

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Wenyan Han ◽  
Guolin Yu ◽  
Tiantian Gong
Keyword(s):  
Optimality Conditions ◽  
Generalized Convexity ◽  
Multiobjective Programming ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Efficient Solutions ◽  
Type I ◽  
Generalized Convex Functions ◽  
Alternative Theorem ◽  
Multiobjective Programming Problem

This note is devoted to the investigation of optimality conditions for robust approximate quasi weak efficient solutions to a nonsmooth uncertain multiobjective programming problem (NUMP). Firstly, under the extended nonsmooth Mangasarian–Fromovitz constrained qualification assumption, the optimality necessary conditions of robust approximate quasi weak efficient solutions are given by using an alternative theorem. Secondly, a class of generalized convex functions is introduced to the problem (NUMP), which is called the pseudoquasi-type-I function, and its existence is illustrated by a concrete example. Finally, under the pseudopseudo-type-I generalized convexity hypothesis, the optimality sufficient conditions for robust approximate quasi weak efficient solutions to the problem (NUMP) are established.

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