Some Theories And Methods For Multiobjective Programming Problems

This dissertation studies mainly theory and methods of multiobjective programming , including penalty function method, optimality conditions and duality of nonsmooth multi-objective fractional programming problems. The main results obtained in this dissertation may be summarized as follows:1. The exponential penalty function method of multiobjective programming problems is studied. Following the classical exponential penalty function method of mathematical programming, the exponential penalty function method of multiobjective programming problems is constructed and convergence of the exponential penalty function method of multiobjective programming problems is proved. When the exponential penalty function method of multiobjective programming problems is utilized for the multiobjective programming problem equalient to the finite minmax multiobjective programming problem, the exponential penalty function method for this class of nondifferentiable multiobjective programming problem is estabished, which is consistent to Maximum Entropy Method [74, 128] for the finite minmax multiobjective programming problem.2. A unified framework for constructing penalty functions of multiobjective programming problems is offered. First, for multiobjective programming problems with inequality constraints, the exponential penalty function method of multiobjective programming problems is derived by the entropy smooth method. Furthermore, when adding a vector separate function produced by the clouse, normal, strict convex function, to the vector Lagrangian function, we develop the vector Lagrangian smooth approach for multiobjective programming problems with inequality constraints. At last, using the vector Lagrangian smooth approach for multiobjective programming problems with inequality constraints, we offer a unified framework for constructing penalty functions of multiobjective programming problems.3. Optimality conditions and duality for three kinds of nonsmooth multiobjective fractional programming problems are considered.(1) For nonsmooth multiobjective fractional programming problems with inequality constraints, we introuced generalized (F, θ, ρ, d) -convex and generalized (F,α, ρ, d) — V-convexconcepts by Clarke subdifferential. Under generalized (F,d,p,d)— convex and generalized (F, a, p, d) — V-convex conditions, optimality conditions of Pareto efficient solutions are established for nonsmooth multiobjective fractional programming problems. Furthermore, the Mond-Weir type duality model, a semiparametric duality model and a parametric duality model are constructed. For every duality model, appropriate duality theorems are proved.(2) A class of multiobjective fractional programming problems are studied, where the involved functions are local Lipschitz and Clarke subdifferentiable. First, under G — (F, p) convexity, the alternative theorem is proved. Moreover, sufficient condition and necessary condition for a properly efficient solution in the sense of Geoffrion are proved.(3) Necessary and sufficient conditions are derived for e-weak efficient solutions of nondifferentiable multiobjective fractional programming problems with nondifferentiable objective functions subject to nondifferentiable convex inequality constraints, linear equality constraints and abstract constraints by e-subdifferential. Based on these, one parametric duality model is constructed and appropriate e-duality theorems are proved.4.0ptimality conditions for multiobjective programming problems having F—convex objective and constraint functions are investigated. An equivalent multiobjective programming problem is constructed by a modification of the objective function. Furthermore, a F—Lagrange function is introduced for a constructed multiobjective programming problem, and a new type of saddle point is introduced. At last, the relations between the new type of saddle point and primal multiobjective programming problem are given.