Abstract
The article is devoted to the development of numerical methods for solving saddle point problems and variational inequalities with simplified requirements for the smoothness conditions of functionals. Recently, some notable methods for optimization problems with strongly monotone operators were proposed. Our focus here is on newly proposed techniques for solving strongly convex-concave saddle point problems. One of the goals of the article is to improve the obtained estimates of the complexity of introduced algorithms by using accelerated methods for solving auxiliary problems. The second focus of the article is introducing an analogue of the boundedness condition for the operator in the case of arbitrary (not necessarily Euclidean) prox structure. We propose an analogue of the Mirror Descent method for solving variational inequalities with such operators, which is optimal in the considered class of problems.
The research in Introduction and Sects. 3,4 is supported by the Ministry of Science and Higher Education of the Russian Federation (Goszadaniye in MIPT, project 075-00337-20-03). The research in Algorithm 3 was partially supported by the grant of the President of Russian Federation for young candidates of sciences (project MK-15.2020.1). The research in Theorem 1 and partially in Sect. 3.1 was supported by the Russian Science Foundation (project 18-71-10044).
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Titov, A.A., Stonyakin, F.S., Alkousa, M.S., Gasnikov, A.V. (2021). Algorithms for Solving Variational Inequalities and Saddle Point Problems with Some Generalizations of Lipschitz Property for Operators. In: Strekalovsky, A., Kochetov, Y., Gruzdeva, T., Orlov, A. (eds) Mathematical Optimization Theory and Operations Research: Recent Trends. MOTOR 2021. Communications in Computer and Information Science, vol 1476. Springer, Cham. https://doi.org/10.1007/978-3-030-86433-0_6
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