An algorithm for convex constrained minimax optimization based on duality

We consider the problem of finding the minimum value of the upper hull of n convex functionals on a Hilbert space, subject to convex constraints. The problem is reformulated as that of finding the minimum of the "worst" convex combination of these functionals, which eventually yields a saddle-point problem. We propose a new algorithm to solve this problem that simplifies the task of updating the dual variables. Simultaneously, the constraints can be dualized by introducing other dual multipliers. Convergence proofs are given and a concrete example shows the practical and computational advantages of the proposed algorithm and approach.