Decomposition Techniques for Bilinear Saddle Point Problems and Variational Inequalities with Affine Monotone Operators on Domains Given by Linear Minimization Oracles

The majority of First Order methods for large-scale convex-concave saddle point problems and variational inequalities with monotone operators are proximal algorithms which at every iteration need to minimize over problem's domain X the sum of a linear form and a strongly convex function. To make such an algorithm practical, X should be proximal-friendly -- admit a strongly convex function with easy to minimize linear perturbations. As a byproduct, X admits a computationally cheap Linear Minimization Oracle (LMO) capable to minimize over X linear forms. There are, however, important situations where a cheap LMO indeed is available, but X is not proximal-friendly, which motivates search for algorithms based solely on LMO's. For smooth convex minimization, there exists a classical LMO-based algorithm -- Conditional Gradient. In contrast, known to us LMO-based techniques for other problems with convex structure (nonsmooth convex minimization, convex-concave saddle point problems, even as simple as bilinear ones, and variational inequalities with monotone operators, even as simple as affine) are quite recent and utilize common approach based on Fenchel-type representations of the associated objectives/vector fields. The goal of this paper is to develop an alternative (and seemingly much simpler) LMO-based decomposition techniques for bilinear saddle point problems and for variational inequalities with affine monotone operators

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 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 Linear Minimization Oracles. The techniques we develope can be viewed as a substantial extension of the proposed in [5] method of nonsmooth convex minimization over an LMO-represented domain