Multi-step algorithms for solving EPs

• Published in: Mathematical modelling and analysis Vol. 23; no. 3; pp. 453 - 472
• Main Authors: Anh, Pham Ngoc, Hieu, Dang Van
• Format: Journal Article
• Language: English
• Published: Vilnius Vilnius Gediminas Technical University 01.01.2018
• Subjects: Algorithms; Convergence; equilibrium problem; Equilibrium selection; extragradient method; Hilbert space; Lipschitz-type continuous; Mathematical research; multi-step method; proximal-like method; Lipschitz-type continuous; multi-step method; extragradient method; proximal-like method; equilibrium problem
• ISSN: 1392-6292; 1648-3510
• DOI: 10.3846/mma.2018.027

The paper introduces and analysizes the convergence of two multi-step proximal-like algorithms for pseudomonotone and Lipschitz-type continuous equilibrium problems in a real Hilbert space. The algorithms are combinations between the multi-step proximal-like method and Mann or Halpern iterations. The weakly and strongly convergent theorems are established with the prior knowledge of two Lipschitz-type continuous constants. Moreover, by choosing two sequences of suitable stepsizes, we also show that the multi-step proximal-like algorithm for strongly pseudomonotone and Lipschitz-type continuous equilibrium problems where the construction of solution approximations and the establishing of its convergence do not require the prior knowledge of strongly pseudomonotone and Lipschitz-type continuous constants of bifunctions. Finally, several numerical examples are reported to illustrate the convergence and the performance of the proposed algorithms over classical extragradient-like algorithms.