An algorithm for solving two-level convex optimization problems

Abstract

This paper provides an algorithm for solving two-level convex optimization problems. The algorithm is based on the subgradient formula for the upper level objective function which is not generally differentiate because of the lower level optimization. It is easy in convex problems to judge whether the upper level objective function is differentiable or not at some point. Hence, if the function is differentiate, the gradient vector is used as a search direction in the algorithm, or, otherwise, a descent direction finding problem, which is a linear programming problem, is formulated and solved. The efficiency of the algorithm is demonstrated by a simple numerical example.