scholarly journals The generalized optimality conditions of multiobjective programming problem in topological vector space

2004 ◽  
Vol 290 (2) ◽  
pp. 363-372 ◽  
Author(s):  
Yuda Hu ◽  
Chen Ling
Keyword(s):  
Vector Space ◽  
Optimality Conditions ◽  
Programming Problem ◽  
Topological Vector Space ◽  
Multiobjective Programming ◽  
Multiobjective Programming Problem

scholarly journals New Optimality Conditions for a Nondifferentiable Fractional Semipreinvex Programming Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Yi-Chou Chen ◽  
Wei-Shih Du
Keyword(s):  
Vector Space ◽  
Optimality Conditions ◽  
Programming Problem ◽  
Topological Vector Space ◽  
Fractional Programming ◽  
Sufficient Conditions ◽  
Continuous Map ◽  
Sufficient Conditions For Optimality ◽  
Necessary And Sufficient ◽  
Locally Convex

We study a nondifferentiable fractional programming problem as follows:(P)minx∈Kf(x)/g(x)subject tox∈K⊆X,  hi(x)≤0,  i=1,2,…,m, whereKis a semiconnected subset in a locally convex topological vector spaceX,f:K→ℝ,g:K→ℝ+andhi:K→ℝ,i=1,2,…,m. Iff,-g, andhi,i=1,2,…,m, are arc-directionally differentiable, semipreinvex maps with respect to a continuous mapγ:[0,1]→K⊆Xsatisfyingγ(0)=0andγ′(0+)∈K, then the necessary and sufficient conditions for optimality of(P)are established.

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 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 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.

Multiobjective Programming

2014 ◽  
pp. 1585-1604
Author(s):  
Minghe Sun
Keyword(s):  
Programming Problem ◽  
Multiobjective Programming ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Objective Functions ◽  
Solution Quality ◽  
Solution Methods ◽  
Multiobjective Programming Problem ◽  
Feasible Solutions ◽  
Multiobjective Programming Problems

Optimization problems with multiple criteria measuring solution quality can be modeled as multiobjective programming problems. Because the objective functions are usually in conflict, there is not a single feasible solution that can optimize all objective functions simultaneously. An optimal solution is one that is most preferred by the decision maker (DM) among all feasible solutions. An optimal solution must be nondominated but a multiobjective programming problem may have, possibly infinitely, many nondominated solutions. Therefore, tradeoffs must be made in searching for an optimal solution. Hence, the DM's preference information is elicited and used when a multiobjective programming problem is solved. The model, concepts and definitions of multiobjective programming are presented and solution methods are briefly discussed. Examples are used to demonstrate the concepts and solution methods. Graphics are used in these examples to facilitate understanding.

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.

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