Weak and strong convergence of proximal penalization and proximal splitting algorithms for two-level hierarchical Ky Fan minimax inequalities

• Published in: Optimization Vol. 64; no. 5; pp. 1285 - 1303
• Main Authors: Chbani, Zaki, Riahi, Hassan
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 04.05.2015; Taylor & Francis LLC
• Subjects: Algorithms; Convergence; Convex analysis; D-proximal penalization algorithm; D-proximal splitting algorithm; Economics; hierarchical Ky Fan minimax inequality; Inequalities; Mathematical analysis; Mathematical functions; Mathematical models; Mathematical problems; Mathematical programming; Minimax technique; Optimization; Splitting; Studies; weak and strong convergence
• ISSN: 0233-1934; 1029-4945
• DOI: 10.1080/02331934.2013.858397

The theory of Ky Fan minimax inequalities provides a powerful general framework for the study of convex programming, variational inequalities and economic equilibrium problems. One of the fundamental methods for finding a solution of Ky Fan minimax inequalities is the proximal point algorithm, where a lot of papers have been dedicated to this subject. In this paper, a general class of two-level hierarchical Ky Fan minimax inequalities is introduced in real Hilbert spaces. For a wide class of Bregman functions, an association of inexact implicit Bregman-penalization proximal and Bregman-splitting proximal algorithms are suggested and analysed. Weak and strong convergences are proved under essentially weaker conditions. We conclude this paper with a hierarchical minimization problem and a numerical example.