scholarly journals Duality for Multitime Multiobjective Ratio Variational Problems on First Order Jet Bundle

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Mihai Postolache
Keyword(s):  
Variational Problems ◽  
Efficient Solutions ◽  
Integral Type ◽  
Duality Theorems ◽  
Jet Bundle ◽  
New Class ◽  
Special Cases ◽  
Multiobjective Variational Problems ◽  
Curvilinear Integral ◽  
Primal And Dual

We consider a new class of multitime multiobjective variational problems of minimizing a vector of quotients of functionals of curvilinear integral type. Based on the efficiency conditions for multitime multiobjective ratio variational problems, we introduce a ratio dual of generalized Mond-Weir-Zalmai type, and under some assumptions of generalized convexity, duality theorems are stated. We prove our weak duality theorem for efficient solutions, showing that the value of the objective function of the primal cannot exceed the value of the dual. Direct and converse duality theorems are stated, underlying the connections between the values of the objective functions of the primal and dual programs. As special cases, duality results of Mond-Weir-Zalmai type for a multitime multiobjective variational problem are obtained. This work further develops our studies in (Pitea and Postolache (2011)).

scholarly journals Duality for multiobjective variational problems under second-order (Ф,ρ)-invexity

Filomat ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 605-615
Author(s):  
Vivek Singh ◽  
I. Ahmad ◽  
S.K. Gupta ◽  
S. Al-Homidan
Keyword(s):  
Variational Problem ◽  
Dual Problem ◽  
Variational Problems ◽  
Efficient Solutions ◽  
Second Order ◽  
Duality Theorems ◽  
Primal Problem ◽  
Duality Relations ◽  
Multiobjective Variational Problems ◽  
Invex Function

The purpose of this article is to introduce the concept of second order (?,?)-invex function for continuous case and apply it to discuss the duality relations for a class of multiobjective variational problem. Weak, strong and strict duality theorems are obtained in order to relate efficient solutions of the primal problem and its second order Mond-Weir type multiobjective variational dual problem using aforesaid assumption. A non-trivial example is also exemplified to show the presence of the proposed class of a function.

scholarly journals Multiobjective duality with ρ−(η,θ)-invexity

2005 ◽  
Vol 2005 (2) ◽  
pp. 175-180 ◽  
Author(s):  
C. Nahak ◽  
S. Nanda
Keyword(s):  
Programming Problem ◽  
Multiobjective Programming ◽  
Efficient Solutions ◽  
Duality Theorems ◽  
Converse Duality ◽  
Dual Problems ◽  
Properly Efficient Solutions ◽  
Multiobjective Programming Problem ◽  
Primal And Dual Problems ◽  
Primal And Dual

Under ρ−(η,θ)-invexity assumptions on the functions involved, weak, strong, and converse duality theorems are proved to relate properly efficient solutions of the primal and dual problems for a multiobjective programming problem.

scholarly journals Mixed type symmetric and self duality for multiobjective variational problems with support functions

2013 ◽  
Vol 23 (3) ◽  
pp. 387-417
Author(s):  
I. Husain ◽  
Rumana Mattoo
Keyword(s):  
Mixed Type ◽  
Variational Problems ◽  
Kernel Functions ◽  
Boundary Values ◽  
Natural Boundary ◽  
Duality Theorems ◽  
Support Functions ◽  
Self Duality ◽  
Multiobjective Nonlinear Programming ◽  
Multiobjective Variational Problems

In this paper, a pair of mixed type symmetric dual multiobjective variational problems containing support functions is formulated. This mixed formulation unifies two existing pairs Wolfe and Mond-Weir type symmetric dual multiobjective variational problems containing support functions. For this pair of mixed type nondifferentiable multiobjective variational problems, various duality theorems are established under convexity-concavity and pseudoconvexity-pseudoconcavity of certain combination of functionals appearing in the formulation. A self duality theorem under additional assumptions on the kernel functions that occur in the problems is validated. A pair of mixed type nondifferentiable multiobjective variational problem with natural boundary values is also formulated to investigate various duality theorems. It is also pointed that our duality theorems can be viewed as dynamic generalizations of the corresponding (static) symmetric and self duality of multiobjective nonlinear programming with support functions.

On the multiobjective control problem

2018 ◽  
Vol 24 (2) ◽  
pp. 223-231
Author(s):  
Promila Kumar ◽  
Bharti Sharma
Keyword(s):  
Control Problem ◽  
Strong Duality ◽  
Efficient Solutions ◽  
Higher Order ◽  
Sufficient Optimality Conditions ◽  
Duality Theorems ◽  
Multiobjective Control ◽  
Primal And Dual Problems ◽  
Primal And Dual

Abstract In this paper, sufficient optimality conditions are established for the multiobjective control problem using efficiency of higher order as a criterion for optimality. The ρ-type 1 invex functionals (taken in pair) of higher order are proposed for the continuous case. Existence of such functionals is confirmed by a number of examples. It is shown with the help of an example that this class is more general than the existing class of functionals. Weak and strong duality theorems are also derived for a mixed dual in order to relate efficient solutions of higher order for primal and dual problems.

scholarly journals Second-order symmetric duality in multiobjective variational problems

2019 ◽  
Vol 29 (3) ◽  
pp. 295-308
Author(s):  
Geeta Sachdev ◽  
Khushboo Verma ◽  
T.R. Gulati
Keyword(s):  
Variational Problems ◽  
Second Order ◽  
Static Case ◽  
Duality Theorems ◽  
Symmetric Duality ◽  
Converse Duality ◽  
Multiobjective Variational Problems ◽  
Second Order Symmetric Duality

In this work, we introduce a pair of multiobjective second-order symmetric dual variational problems. Weak, strong, and converse duality theorems for this pair are established under the assumption of ?-bonvexity/?-pseudobonvexity. At the end, the static case of our problems has also been discussed.

scholarly journals Sufficient Efficiency Conditions for Vector Ratio Problem on the Second-Order Jet Bundle

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Ariana Pitea
Keyword(s):  
Optimality Conditions ◽  
Mechanical Engineering ◽  
Optimization Theory ◽  
Necessary Optimality Conditions ◽  
Variational Problems ◽  
Second Order ◽  
Mechanical Work ◽  
Jet Bundle ◽  
Multiobjective Variational Problems ◽  
Efficiency Conditions

Motivated by its possible applications in mechanics and mechanical engineering, in our previous published work (Pitea and Postolache, 2011), we initiated an optimization theory for the second-order jet bundle. We considered the problem of minimization of vectors of curvilinear functionals (well known as mechanical work), thought as multitime multiobjective variational problems, subject to PDE and/or PDI constraints. Within this framework, we introduced necessary optimality conditions. As natural continuation of these results, the present work introduces a study of sufficient efficiency conditions.

ϵ-Efficient solutions in semi-infinite multiobjective optimization

2018 ◽  
Vol 52 (4-5) ◽  
pp. 1397-1410
Author(s):  
Tatiana Shitkovskaya ◽  
Do Sang Kim
Keyword(s):  
Multiobjective Optimization ◽  
Nonsmooth Analysis ◽  
Constraint Qualification ◽  
Dual Problem ◽  
Optimization Problems ◽  
Efficient Solutions ◽  
Duality Theorems ◽  
Multiobjective Optimization Problems ◽  
Special Cases ◽  
Scalarization Method

In this paper we apply some tools of nonsmooth analysis and scalarization method due to Chankong–Haimes to find ϵ-efficient solutions of semi-infinite multiobjective optimization problems (MP). We establish ϵ-optimality conditions of Karush–Kuhn–Tucker (KKT) type under Farkas–Minkowski (FM) constraint qualification by using ϵ-subdifferential concept. In addition we propose mixed type dual problem (including dual problems of Wolfe and Mond–Weir types as special cases) for ϵ-efficient solutions and investigate relationship between mentioned (MP) and its dual problem as well as establish several ϵ-duality theorems.

scholarly journals Efficiency and duality for multiobjective fractional variational problems with (ρ,b) - quasiinvexity

2009 ◽  
Vol 19 (1) ◽  
pp. 85-99 ◽  
Author(s):  
Ştefan Mititelu ◽  
I.M. Stancu-Minasian
Keyword(s):  
Necessary Conditions ◽  
Inequality Constraints ◽  
Variational Problems ◽  
Efficient Solutions ◽  
Parametric Approach ◽  
Duality Theorems ◽  
Normal Efficiency ◽  
Fractional Variational Problems ◽  
Equality And Inequality Constraints ◽  
Nonlinear Equality

The necessary conditions for (normal) efficient solutions to a class of multi-objective fractional variational problems (MFP) with nonlinear equality and inequality constraints are established using a parametric approach to relate efficient solutions of a fractional problem and a non-fractional problem. Based on these normal efficiency criteria a Mond-Weir type dual is formulated and appropriate duality theorems are proved assuming (?,b) - quasi-invexity of the functions involved.

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