A class of nonconvex functions and pre-variational inequalities

A class of nonconvex functions is introduced, called semi-preinvex function, which includes the classes of preinvex functions and arc-connected convex functions. The Fritz-John conditions of the mathematical programming problem are derived for these kinds of functions. The pre-variational inequality is given as a necessary condition and also a sufficient condition for a mathematical programming for invex functions. The Type I function related to unconstrained problems is given as an equivalent form of the pre-variational inequality. Existence theorems for the solution of the pre-variational inequality are also proved.