A Strong Convergence Theorem for an Iterative Method for Finding Zeros of Maximal Monotone Maps with Applications to Convex Minimization and Variational Inequality Problems

• Published in: Proceedings of the Edinburgh Mathematical Society Vol. 62; no. 1; pp. 241 - 257
• Main Authors: Chidume, C. E., Uba, M. O., Uzochukwu, M. I., Otubo, E. E., Idu, K. O.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.02.2019
• Subjects: Algorithms; Banach spaces; Computational geometry; Convergence; Convexity; Optimization; Portfolio management; Theorems; monotone mapping; 47J20; Primary 47H09; 47J05; maximal monotone mappings; 47H05; 47HJ25; strong convergence; Secondary 47H14
• ISSN: 0013-0915; 1464-3839
• DOI: 10.1017/S0013091518000366

Let E be a uniformly convex and uniformly smooth real Banach space, and let E* be its dual. Let A : E → 2E* be a bounded maximal monotone map. Assume that A−1(0) ≠ Ø. A new iterative sequence is constructed which converges strongly to an element of A−1(0). The theorem proved complements results obtained on strong convergence of the proximal point algorithm for approximating an element of A−1(0) (assuming existence) and also resolves an important open question. Furthermore, this result is applied to convex optimization problems and to variational inequality problems. These results are achieved by combining a theorem of Reich on the strong convergence of the resolvent of maximal monotone mappings in a uniformly smooth real Banach space and new geometric properties of uniformly convex and uniformly smooth real Banach spaces introduced by Alber, with a technique of proof which is also of independent interest.