The Optimality Conditions Of A Class Nonsmooth Multiobjective Optimization Problems

Multiobjective optimization is an important branch of optimization and it is a widely used in engineering technology, production management and national defense construction, etc. In recent years, many scholars on multi-objective programming and the application of the theory of optimization have done a lot of research, Many scholars of multi-objective optimization problem with untiring discussion and research, makes the multi-objective optimization theory and algorithm developed rapidly.However, the majority of function model in actual application is non-smooth situation, before 1960s, because of the tool of mathematical analysis limit, always unconsciously assumed functions involved are smooth. With the development of convex analysis, many scholars begin to study the theory of non-smooth, so the non-smooth theory made great breakthroughs, the theory and convex function is extended to the generalized convex function of various meanings. Since the proposed invariant Hanson B-convex function, F-convex function, F-ρ convex function. (C,α,ρ,d) convex function have been proposed.on the basis of existing research results, we propose B-(C,α) and B-(C,α)-Itype of generalized convex function classes, namely(C,α,ρ,d) generalized convex function is a special case of B-(C,α) generalized convex functions, in addition to the B-(C,α) generalized convex function and B-(C,α)-I type of generalized convex function classes weaken the concept of convex function, to a certain extent, extension of the convex function.This paper structure as follows:the first chapter presents a non-smooth programming background and the developing situation; the second chapter give the base-knowledge of pareto efficient solution, weak pareto efficient solution, and development of the generalized convexity function; the third chapter in view of the objective function and the restraint function based B-(C,α)-I type function, introduce a new generalized class of B-(C, α)-I type, pseud B-(C, α)-I type, quasi B-(C, α)-I type and quasi-pseudo B-(C,α)-I type univex. At last, discuss and prove optimality condition of the programming, discuss the Mond-Weir dual problem and study the dual theory. The fourth chapter based on the B-(C,α)-Itype of generalized convex functions, is discussed for a class of non-smooth multiobjective fractional programming problems, by adopting the method of constructing auxiliary function, combined with certain constraints specifications, the fraction problem is transformed into equivalent general planning problem, gives the sufficient conditions of optimality of this kind of problem and duality theorem.