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

The standard algorithms for solving large-scale convex–concave saddle point problems, or, more generally, variational inequalities with monotone operators, are proximal type algorithmswhich 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 cheapLinear Minimization Oracle (LMO) able tominimize 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. Mathematics Subject Classification 65K15 · 90C25 · 90C47 · 68T10 Research of the first author was supported by the CNRS-Mastodons project GARGANTUA, and the LabEx PERSYVAL-Lab (ANR-11-LABX-0025). Research of the second author was supported by the NSF grants CMMI 1232623 and CCF 1415498. A. Juditsky (B) LJK, Université Grenoble Alpes, B.P. 53, 38041 Grenoble Cedex 9, France e-mail: [email protected] A. Nemirovski Georgia Institute of Technology, Atlanta, GA 30332, USA e-mail: [email protected]