Solving variational inequalities with monotone operators on domains given by Linear Minimization Oracles

Abstract

The standard algorithms for solving large-scale convex---concave saddle point problems, or, more generally, variational inequalities with monotone operators, are proximal type algorithms which at every iteration need to compute a prox-mapping, that is, to minimize over problem's domain X the sum of a linear form and the specific convex distance-generating function underlying the algorithms in question. (Relative) computational simplicity of prox-mappings, which is the standard requirement when implementing proximal algorithms, clearly implies the possibility to equip X with a relatively computationally cheap Linear Minimization Oracle (LMO) able to minimize over X linear forms. There are, however, important situations where a cheap LMO indeed is available, but where no proximal setup with easy-to-compute prox-mappings is known. This fact motivates our goal in this paper, which is to develop techniques for solving variational inequalities with monotone operators on domains given by LMO. The techniques we discuss can be viewed as a substantial extension of the proposed in Cox et al. (Math Program Ser B 148(1---2):143---180, 2014) method of nonsmooth convex minimization over an LMO-represented domain.