Generalized Proximal Method for Efficient Solutions in Vector Optimization

• Published in: Numerical functional analysis and optimization Vol. 32; no. 8; pp. 843 - 857
• Main Author: Chuong, T. D.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Taylor & Francis Group 01.08.2011; Taylor & Francis
• Subjects: Algorithms; Banach space; Calculus of variations and optimal control; Efficient solution; Exact sciences and technology; Functional analysis; Generalized proximal method; Mapping; Mathematical analysis; Mathematical models; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in mathematical programming, optimization and calculus of variations; Numerical methods in optimization and calculus of variations; Optimization; Sciences and techniques of general use; Uniformly convex; Uniformly smooth; Vector optimization; Vectors (mathematics); Generalized proximal method; 65K05; Efficient solution; Uniformly smooth; Optimization method; Banach space; Algorithm; Weak convergence; Numerical analysis; 49J52; 90C25; Ordering; Variational calculus; Uniformly convex; Vector optimization; Mathematical programming; 49M99
• ISSN: 0163-0563; 1532-2467
• DOI: 10.1080/01630563.2011.587072

This article is devoted to developing the generalized proximal algorithm of finding efficient solutions to the vector optimization problem for a mapping from a uniformly convex and uniformly smooth Banach space to a real Banach space with respect to the partial order induced by a pointed closed convex cone. In contrast to most published literature on this subject, our algorithm does not depend on the nonemptiness of ordering cone of the space under consideration and deals with finding efficient solutions of the vector optimization problem in question. We prove that under some suitable conditions the sequence generated by our method weakly converges to an efficient solution of this problem.