Optimality And Duality For Certain Generalized Convex Programming Problems

In this paper, the definitions of certain generalized convex functions are presentedby means the concepts of local starshaped E-convex set, semilocal convex function andb-convex function, those are, semilocalE_b-convex function, semilocalE b-quasiconvexfunction, semilocalE_b-pseudoconvex function, semilocalE_b-strict convex function.The relationship between these generalized convex functions and other convex functionsare researched. The optimality conditions and duality results for these generalized convexprogramming problems and multi-objective semi-infinite programming problems arestudied. The main content of this paper includes several sides as below:(1) The semilocalE b-convex function and some generalized convex functions aredefined by using the concepts of local starshaped E-convex set, semilocal convexfunction and b-convex function. Furthermore, the relationship between these generalizedconvex functions and some other convex functions are discussed;(2) Optimality and Lagrange type duality theory for these generalized convexprogramming problems are studied involving these generalized convexity;(3) The optimality conditions and Mond Weirtype duality theory formulti-objective semi-infinite programming problems are given and proofed involvingthese semilocalE_b-convex functions;(4) The optimality conditions and duality results for multi-objective semi-infinitefractional programming problems are presented involving these semilocalE_b-convexfunctions.In all, in this paper, some classes of more generalized convex functions are given.The optimality conditions and duality results for generalized convex programming,multi-objective semi-infinite programming and multi-objective semi-infinite fractionalprogramming are obtained based on these generalized convexity, which enriched the theory of semi-infinite programming problem.