2013/12/04 by Anatoli Juditsky, Arkadi Nemirovski, Juditsky, Anatoli +1 · 4 citations
Computer Science · Engineering · Mathematics · #Advanced Optimization Algorithms Research #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis #Sparse and Compressive Sensing Techniques
paper · doi:10.48550/arxiv.1312.1073
openalex publication_date 2013/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.