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Article Dans Une Revue Journal of Optimization Theory and Applications Année : 2017

Decomposition Techniques for Bilinear Saddle Point Problems and Variational Inequalities with Affine Monotone Operators

Résumé

The majority of first-order methods for large-scale convex–concave saddle point problems and variational inequalities with monotone operators are proximal algorithms. To make such an algorithm practical, the problem’s domain should be proximal-friendly—admit a strongly convex function with easy to minimize linear perturbations. As a by-product, this domain admits a computationally cheap linear minimization oracle (LMO) capable to minimize linear forms. There are, however, important situations where a cheap LMO indeed is available, but the problem domain is not proximal-friendly, which motivates search for algorithms based solely on LMO. For smooth convex minimization, there exists a classical algorithm using LMO—conditional gradient. In contrast, known to us similar 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 was to develop alternative (and seemingly much simpler) decomposition techniques based on LMO for bilinear saddle point problems and for variational inequalities with affine monotone operators.

Dates et versions

hal-01336002 , version 1 (22-06-2016)

Identifiants

Citer

Bruce Cox, Anatoli B. Juditsky, Arkadii S. Nemirovskii. Decomposition Techniques for Bilinear Saddle Point Problems and Variational Inequalities with Affine Monotone Operators. Journal of Optimization Theory and Applications, 2017, 172 (2), pp.402-435. ⟨10.1007/s10957-016-0949-3⟩. ⟨hal-01336002⟩
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