Sufficiency and duality in differentiable multiobjective programming involving generalized type I functions

AbstractIn this paper, new classes of generalized (F,α,ρ,d)-type I functions are introduced for differentiable multiobjective programming. Based upon these generalized functions, first, we obtain several sufficient optimality conditions for feasible solution to be an efficient or weak efficient solution. Second, we prove weak and strong duality theorems for mixed type duality

Optimality conditions and duality for nondifferentiable multiobjective programming problems involving d-r-type I functions

AbstractIn this paper, new classes of nondifferentiable functions constituting multiobjective programming problems are introduced. Namely, the classes of d-r-type I objective and constraint functions and, moreover, the various classes of generalized d-r-type I objective and constraint functions are defined for directionally differentiable multiobjective programming problems. Sufficient optimality conditions and various Mond–Weir duality results are proved for nondifferentiable multiobjective programming problems involving functions of such type. Finally, it is showed that the introduced d-r-type I notion with r≠0 is not a sufficient condition for Wolfe weak duality to hold. These results are illustrated in the paper by suitable examples

Nonsmooth multiobjective optimization using limiting subdifferentials

AbstractIn this study, using the properties of limiting subdifferentials in nonsmooth analysis and regarding a separation theorem, some weak Pareto-optimality (necessary and sufficient) conditions for nonsmooth multiobjective optimization problems are proved

An n-order (F,a,p,d)- Convex Function and Duality Problem

A class of n-order (F,a,p, d)-convex function and their generalization on functions is introduced. Using the assumption on the functions involved,weak, strong ,and converse duality theorems are established for the n-order dual proble

On nonsmooth multiobjective fractional programming problems involving (p, r)− ρ −(η ,θ)- invex functions

A class of multiobjective fractional programming problems (MFP) is considered where the involved functions are locally Lipschitz. In order to deduce our main results, we introduce the definition of (p,r)−ρ −(η,θ)-invex class about the Clarke generalized gradient. Under the above invexity assumption, sufficient conditions for optimality are given. Finally, three types of dual problems corresponding to (MFP) are formulated, and appropriate dual theorems are proved

Duality for MultiobjectiveB-vex Programming Involvingn-Set Functions

AbstractIn the present paper we consider a class of multiobjectiveB-vex programming problems involving differentiableB-vexn-set functions and establish duality results in terms of properly efficient solutions. Further, we relate the problem to a certain saddle point of a Lagrangian and show multiobjective fractional program as a special case of the main problem

Exact Penalization and Necessary Optimality Conditions for Multiobjective Optimization Problems with Equilibrium Constraints

A calmness condition for a general multiobjective optimization problem with equilibrium constraints is proposed. Some exact penalization properties for two classes of multiobjective penalty problems are established and shown to be equivalent to the calmness condition. Subsequently, a Mordukhovich stationary necessary optimality condition based on the exact penalization results is obtained. Moreover, some applications to a multiobjective optimization problem with complementarity constraints and a multiobjective optimization problem with weak vector variational inequality constraints are given

An Interactive Fuzzy Satisficing Method for Multiobjective Nonlinear Programming Problems with Fuzzy Parameters

This paper presents an interactive fuzzy satisficing method for multiobjective nonlinear programming problems with fuzzy parameters. The fuzzy parameters in the objective functions and the constraints are characterized by the fuzzy numbers. On the basis of the alpha-level sets of the fuzzy numbers, the concept of alpha-multiobjective nonlinear programming and alpha-Pareto optimality is introduced. Through the interaction with the decision maker (DM), the fuzzy goals of the DM for each of the objective functions in alpha-multiobjective nonlinear programming are quantified by eliciting the corresponding membership functions. After determining the membership functions, in order to generate a candidate for the satisficing solution which is also alpha-Pareto optimal, if the DM specifies the degree alpha of the alpha-level sets and the reference membership values, the augmented minimax problem is solved and the DM is supplied with the corresponding alpha-Pareto optimal solution together with the trade-off rates among the values of the membership functions and the degree alpha. Then by considering the current values of the membership functions and as well as the trade-off rates, the DM responds by updating his reference membership values and/or the degree alpha. In this way the satisficing solution for the DM can be derived efficiently from among an alpha-Pareto optimal solution set. Based on the proposed method, a time-sharing computer program is written and an illustrative numerical example is demonstrated along with the corresponding computer outputs