Approximate Optimality Conditions for Composite Convex Optimization Problems

The purpose of this paper is to study the approximate optimality condition for composite convex optimization problems with a cone-convex system in locally convex spaces, where all functions involved are not necessarily lower semi-continuous. By using the properties of the epigraph of conjugate functions, we introduce a new regularity condition and give its equivalent characterizations. Under this new regularity condition, we derive necessary and sufficient optimality conditions of ε-optimal solutions for the composite convex optimization problem. As applications of our results, we derive approximate optimality conditions to cone-convex optimization problems. Our results extend or cover many known results in the literature. © 2016, Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag Berlin Heidelberg.