An accelerated non-Euclidean hybrid proximal extragradient-type algorithm for convex–concave saddle-point problems

Abstract

This paper describes an accelerated HPE-type method based on general Bregman distances for solving convex–concave saddle-point (SP) problems. The algorithm is a special instance of a non-Euclidean hybrid proximal extragradient framework introduced by Svaiter and Solodov [An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions, Math. Oper. Res. 25(2) (2000), pp. 214–230] where the prox sub-inclusions are solved using an accelerated gradient method. It generalizes the accelerated HPE algorithm presented in He and Monteiro [An accelerated HPE-type algorithm for a class of composite convex–concave saddle-point problems, SIAM J. Optim. 26 (2016), pp. 29–56] in two ways, namely: (a) it deals with general monotone SP problems instead of bilinear structured SPs and (b) it is based on general Bregman distances instead of the Euclidean one. Similar to the algorithm of He and Monteiro [An accelerated HPE-type algorithm for a class of composite convex–concave saddle-point problems, SIAM J. Optim. 26 (2016), pp. 29–56], it has the advantage that it works for any constant choice of proximal stepsize. Moreover, a suitable choice of the stepsize yields a method with the best known iteration-complexity for solving monotone SP problems. Computational results show that the new method is superior to Nesterov's [Smooth minimization of non-smooth functions, Math. Program. 103(1) (2005), pp. 127–152] smoothing scheme.