Convergence of Optimization Problems

In this paper we consider a general optimization problem (OP) and study the convergence and approximation of optimal values and optimal solutions to changes in the cost function and the set of feasible solutions. We consider the convergence optimization problems under the familiar notion of uniform convergence. We do not assume the convexity of the functions involved. Instead we consider a class of functions whose directional derivatives are convex. They are known as locally convex functions or following Craven and Mond nearly convex functions. We given necessary preliminaries and we prove that a sequence of locally convex optimization problems converge to a locally convex problem. We also prove that uniform convergence of locally convex optimization problems implies epi-graph convergence of the problems. Even though for simplicity we have taken locally convex functions, the results given here can be proved for locally Lipchitz functions also.