Mathematical programming with Semilocally Subconvex functions over cones

In this paper, we introduce another generalization of semilocally convex functions over cones, called conesemilocallysubconvex function (C-slsb), and compare it with other generalizations of convex functions through examples.Further, using its properties we establish a theorem of the alternatives for these functions. Then we investigate the optimalsolutions of the mathematical programming problem (MP) over cones using these functions, directional derivatives, andthe alternative theorem. Investigation of optimal solutions of (MP) is done by deriving optimality and duality results forsemilocally subconvex mathematical programming problems over cones (MP).