Beyond local optimality conditions: the case of maximizing a convex function

In this paper, we design an algorithm for maximizing a convex function over a convex feasible set. The algorithm consists of two phases: in phase 1 a feasible solution is obtained that is used as an initial starting point in phase 2. In the latter, a biconvex problem equivalent to the original problem is solved by employing an alternating direction method. The main contribution of the paper is connected to the first phase. We identify a kind of global optimality condition which says that “The maximizer of a convex objective function is the furthest point from the minimizer”. Using this principle, we develop several ways to compute this ‘furthest point’, focusing on methods that lead to computationally efficient algorithms. The performance of the overall algorithm is tested on a wide variety of problems, demonstrating its efficiency.