Convergence of a Two-Stage Proximal Algorithm for the Equilibrium Problem in Hadamard Spaces
- PDF / 139,804 Bytes
- 9 Pages / 594 x 792 pts Page_size
- 113 Downloads / 196 Views
CONVERGENCE OF A TWO-STAGE PROXIMAL ALGORITHM FOR THE EQUILIBRIUM PROBLEM IN HADAMARD SPACES* Ya. I. Vedel,1† G. V. Sandrakov,1‡ V. V. Semenov,1†† and L. M. Chabak2
UDC 517.988
Abstract. An iterative two-stage proximal algorithm for approximate solution of equilibrium problems in Hadamard spaces is considered. This algorithm is an analog of the already studied two-stage algorithm for equilibrium problems in a Hilbert space. For Lipschitz-type pseudo-monotone bifunctions, a theorem on the weak convergence of sequences generated by the algorithm is proved. Keywords: Hadamard space, equilibrium problem, pseudo-monotonicity, two-stage algorithm, convergence. INTRODUCTION Analysis of equilibrium problems (Ky Fan inequalities, equilibrium programming problems) [1–12] is an important and rather popular field in the modern applied nonlinear analysis. Of interest is creation of the theory and algorithms for solution of mathematical programming problems in Hadamard metric spaces (also known as CAT (0)-spaces), which is due to problems of mathematical biology and machine learning. A strong motivation for analyzing these problems is also the possibility to present some nonconvex problems as convex ones (more precisely, geodesically convex) in the space with specially selected Riemannian metrics [8]. In the present paper, we will continue the studies performed in [10] and propose a two-stage proximal algorithm for approximate solution of equilibrium problems in Hadamard spaces. The algorithm is an analog of two-stage algorithms for variational inequalities and equilibrium problems in a Hilbert space or in an finite-dimensional linear normalized space with Bregman divergence, analyzed earlier in [9, 10]. For pseudo-monotonic bifunctions of Lipschitz type, we will prove the theorem about weak convergence (D -convergence) of the sequences generated by the algorithm. A SURVEY OF THE AVAILABLE RESULTS Classical equilibrium problems have the form [3]: find x ÎÑ : F ( x, y) ³ 0 "y ÎÑ ,
(1)
*
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 number 0219U008403) and by the National Academy of Sciences of Ukraine (project “New methods for the analysis of correctness and solution of discrete optimization problems, variational inequalities, and their application,” state registration number 0119U101608). 1
Taras Shevchenko National University of Kyiv, Kyiv, Ukraine, †[email protected]; ‡[email protected]; [email protected]. 2State University of Infrastructural Technologies, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 5, September–October, 2020, pp. 115–125. Original article submitted November 15, 2019. ††
784
1060-0396/20/5605-0784 ©2020 Springer Science+Business Media, LLC
where Ñ is a nonempty subset of Hilbert space H ; F : C ´ C ® Å is a function (bifunction) such that F ( x, x ) = 0 "x ÎÑ . Mathematical programming p
Data Loading...
