Approximate proximal methods in vector optimization |
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Authors: | Lu-Chuan Ceng Jen-Chih Yao |
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Institution: | 1. Department of Mathematics, Shanghai Normal University, Shanghai 200234, China;2. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan |
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Abstract: | This paper studies the vector optimization problem of finding weakly efficient points for mappings in a Banach space Y, with respect to the partial order induced by a closed, convex, and pointed cone C ⊂ Y with a nonempty interior. The proximal method in vector optimization is extended to develop an approximate proximal method for this problem by virtue of the approximate proximal point method for finding a root of a maximal monotone operator. In this approximate proximal method, the subproblems consist of finding weakly efficient points for suitable regularizations of the original mapping. We present both an absolute and a relative version, in which the subproblems are solved only approximately. Weak convergence of the generated sequence to a weak efficient point is established. In addition, we also discuss an extension to Bregman-function-based proximal algorithms for finding weakly efficient points for mappings. |
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Keywords: | Vector optimization Proximal point Inexact algorithm Banach space |
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