A Class of Semilocal -Preinvex Functions and Its Applications in Nonlinear Programming

A kind of generalized convex set, called as local star-shaped -invex set with respect to , is presented, and some of its important characterizations are derived. Based on this concept, a new class of functions, named as semilocal -preinvex functions, which is a generalization of semi--preinvex functions and semilocal -convex functions, is introduced. Simultaneously, some of its basic properties are discussed. Furthermore, as its applications, some optimality conditions and duality results are established for a nonlinear programming.