An Adaptive Two-Stage Proximal Algorithm for Equilibrium Problems in Hadamard Spaces
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AN ADAPTIVE TWO-STAGE PROXIMAL ALGORITHM FOR EQUILIBRIUM PROBLEMS IN HADAMARD SPACES* Ya. I. Vedel,1† G. V. Sandrakov,1‡ and V. V. Semenov1††
UDC 517.988
Abstract. Equilibrium problems in Hadamard metric spaces are considered in the paper. For approximate solution of problems, a new iterative adaptive two-stage proximal algorithm is proposed and analyzed. In contrast to the previously used rules for choosing the step size, the proposed algorithm does not calculate bifunction values at additional points and does not require knowledge of the value of bifunction’s Lipschitz constants. For pseudo-monotone bifunctions of Lipschitz type, the theorem on weak convergence of the sequences generated by the algorithm is proved. It is shown that the proposed algorithm is applicable to pseudo-monotone variational inequalities in Hilbert spaces. Keywords: Hadamard space, equilibrium problem, pseudo-monotonicity, two-stage proximal algorithm, adaptivity, convergence. INTRODUCTION Analysis of equilibrium problems is a well-known field in the modern applied nonlinear analysis. To solve equilibrium problems in a Hilbert space, a number of publications proposed the two-stage proximal algorithm. Equilibrium problems in Hadamard metric spaces has become of interest, in particular, an analog of the two-stage proximal algorithm has been studied. In the paper, we propose a new adaptive two-stage proximal algorithm for approximate solution of equilibrium problems in Hadamard metric spaces. Unlike the algorithm applied before, this algorithm does not calculate the values of the bifunction at additional points and does not require Lipschitz constants of the bifunction to be known. For pseudo-monotone bifunctions of Lipschitz type, we will prove the theorem about weak convergence of the sequences generated by the algorithm. We will show that the proposed algorithm is applicable to pseudo-monotone variational inequalities in Hilbert spaces. AN OVERVIEW OF THE AVAILABLE RESULTS One of the fields in the modern applied nonlinear analysis is equilibrium problems (Ky Fan inequalities, equilibrium programming problems) of the following form [1–13]: find x ÎÑ : F ( x, y) ³ 0 "y ÎÑ ,
(1)
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The study was financially supported by the Ministry of Education and Science of Ukraine (project “Mathematical modeling and optimization of dynamic systems for defense, medicine, and ecology,” state registration # 0219U008403) and National Academy of Sciences of Ukraine (project “New methods of correctness analysis and solution of discrete optimization problems, variational inequalities, and their application,” state registration # 0119U101608).. 1
Taras Shevchenko National University of Kyiv, Kyiv, Ukraine, †[email protected]; ‡[email protected]; [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 6, November–December, 2020, pp. 136–148. Original article submitted January 22, 2020.
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1060-0396/20/5606-0978 ©2020 Springer Science+Business Media, LLC
where Ñ is a nonempty subset of the Hilbert space H ; F : C ´ C ® R is
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