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

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