Optimality Conditions And Duality Of Programming Problems On The Generalized Convexity

Optimality conditions which is necessary conditions and sufficiency conditions forthe existence of the optimal solution under some kind of meaning and the duality theoryare important parts of the optimization theory, and they have the important theoreticalsignificance and application value. Some important results of the optimality conditionsand dual theory for multi-objective programming and multi-objective fractionalprogramming are directed at the convex and generalized convex programming.Convexity in optimality and dual theory of mathematical programming plays a veryimportant role, so the convex analysis has become the important basis for the subjects ofmathematical programming and optimization theory, but a lot of functions arenon-convex functions actually, therefore convex functions have been promoted, and allsorts of generalized convex functions are obtained. Now many scholars have studiedoptimality conditions and duality theory of various kinds of optimization problemsunder various kinds of generalized convexity conditions, and research on differentiablemulti-objective programming problems is relatively mature, some results of nonsmoothmulti-objective programming problems are obtained under convexity and generalizedconvexity. The main contents of this paper are shown below:For the duality of multi-objective programming problems, the Wolfe vectorduality、Mond-Weir type vector duality and mixed type duality are discussed underthe (F,α,ρ,d)-convexity and generalized (F,α,ρ,d)-convexity which are based onthe F-convexity,-convexity and F,-convexity. The convexity conditions of theweak duality theorems are weakened, and under the weaker convexity conditionsincluding (F,α,ρ,d)-qusiconvexity,(F,α,ρ,d)-pseudoconvexity and weak(F,α,ρ,d)-qusiconvexity, the corresponding weak duality theorems are given andproved. The applied scope of the multi-objective programming is extended byweakening the convexity conditions.Some assumptions for the objective functions and constraint functions are givenunder the conditions of F,-convex,(F,α,ρ,d)-convex, and generalized (F,α,ρ,d)-convex, which are based on the F-convex,-convex and F,-convex,the optimality conditions and dual theory of three kinds of multi-objective fractionalprogramming problems are discussed, and the sufficiency of Kuhn-Tucker optimalityconditions and appropriate duality theorem are proved.In order to discuss the multi-objective fractional programming problem withrespect to locally Lipschitz function, generalized nondifferentiable (F,α,ρ,d)-convexfunctions are given based on the definitions of generalized Clarke gradient andnondifferentiable (F,α,ρ,d)-convex. Under the assumption of generalizednondifferentiable convexity，optimality conditions for a class of nondifferentiablemulti-objective fractional programming problem are obtained.