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Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization

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Abstract

This paper studies a general vector optimization problem of finding weakly efficient points for mappings from Hilbert spaces to arbitrary Banach spaces, where the latter are partially ordered by some closed, convex, and pointed cones with nonempty interiors. To find solutions of this vector optimization problem, we introduce an auxiliary variational inequality problem for a monotone and Lipschitz continuous mapping. The approximate proximal method in vector optimization is extended to develop a hybrid approximate proximal method for the general vector optimization problem under consideration by combining an extragradient method to find a solution of the variational inequality problem and an approximate proximal point method for finding a root of a maximal monotone operator. In this hybrid approximate proximal method, the subproblems consist of finding approximate solutions to the variational inequality problem for monotone and Lipschitz continuous mapping, and then finding weakly efficient points for a suitable regularization of the original mapping. We present both absolute and relative versions of our hybrid algorithm in which the subproblems are solved only approximately. The weak convergence of the generated sequence to a weak efficient point is established under quite mild assumptions. In addition, we develop some extensions of our hybrid algorithms for vector optimization by using Bregman-type functions.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    L. C. Ceng

  2. Scientific Computing Key Laboratory of Shanghai Universities, Shanghai, China

    L. C. Ceng

  3. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    B. S. Mordukhovich

  4. Department of Applied Mathematics, National Sun Yat-sen University, 804, Kaohsiung, Taiwan

    J. C. Yao

Authors
  1. L. C. Ceng
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  2. B. S. Mordukhovich
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  3. J. C. Yao
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Corresponding author

Correspondence to J. C. Yao.

Additional information

Communicated by F. Giannessi.

L.C. Ceng research was partially supported by the Leading Academic Discipline Project of Shanghai Normal University (DZL707), Innovation Program of Shanghai Municipal Education Commission Grant (09ZZ133), National Science Foundation of China (10771141), PhD Program Foundation of Ministry of Education of China (20070270004), Science and Technology Commission of Shanghai Municipality Grant (075105118), and Shanghai Leading Academic Discipline Project (S30405).

B.S. Mordukhovich research was partially supported by the USA National Science Foundation under Grant DMS-0603896.

J.C. Yao research was partially supported by Grant NSC 98-2923-E-110-003-MY3.

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Ceng, L.C., Mordukhovich, B.S. & Yao, J.C. Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization. J Optim Theory Appl 146, 267–303 (2010). https://doi.org/10.1007/s10957-010-9667-4

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  • DOI: https://doi.org/10.1007/s10957-010-9667-4

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