An approximate proximal-extragradient type method for monotone variational inequalities

• Published in: Journal of mathematical analysis and applications Vol. 300; no. 2; pp. 362 - 374
• Main Authors: He, Bing-sheng, Yang, Zhen-hua, Yuan, Xiao-ming
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 15.12.2004; Elsevier
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Calculus of variations and optimal control; Computer science; control theory; systems; Exact sciences and technology; Inexact proximal point algorithm; Mathematical analysis; Mathematics; Monotone variational inequality; Sciences and techniques of general use; Theoretical computing; Monotone variational inequality; Inexact proximal point algorithm; Monotone inequality; Inexact algorithm; Proximal point method; Mathematical analysis; Variational inequality
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2004.04.068

Proximal point algorithms (PPA) are attractive methods for monotone variational inequalities. The approximate versions of PPA are more applicable in practice. A modified approximate proximal point algorithm (APPA) presented by Solodov and Svaiter [Math. Programming, Ser. B 88 (2000) 371–389] relaxes the inexactness criterion significantly. This paper presents an extended version of Solodov–Svaiter's APPA. Building the direction from current iterate to the new iterate obtained by Solodov–Svaiter's APPA, the proposed method improves the profit at each iteration by choosing the optimal step length along this direction. In addition, the inexactness restriction is relaxed further. Numerical example indicates the improvement of the proposed method.