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Hybrid approximate proximal method with auxiliary variational inequality for vector optimization. (English) Zbl 1251.90388

Considering general vector optimization problems in infinite dimensions, the authors combine ideas from proximal point methods in vector optimization with iterative gradient schemes for finding approximate solutions of auxiliary variational inequalities into a hybrid approximate proximal point method (HAPM). After reviewing the recent literature on iterative algorithms for vector optimization and related problems, the authors first present an absolute version of the HAPM method and prove convergence results to a weak efficient minimizer of the considered vector optimization problem (and simultaneously to a solution of an auxiliary variational inequality) under rather mild assumptions. Based on a Bregman function extension of the absolute version of the HAPM method, the authors subsequently suggest a relative version of this approach and derive corresponding convergence results.

MSC:

90C48 Programming in abstract spaces
90C56 Derivative-free methods and methods using generalized derivatives
49K99 Optimality conditions
90C29 Multi-objective and goal programming
90C59 Approximation methods and heuristics in mathematical programming
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