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Exceptional Families of Elements for Optimization Problems in Reflexive Banach Spaces with Applications

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Abstract

In this paper, we propose a new notion of ‘exceptional family of elements’ for convex optimization problems. By employing the notion of ‘exceptional family of elements’, we establish some existence results for convex optimization problem in reflexive Banach spaces. We show that the nonexistence of an exceptional family of elements is a sufficient and necessary condition for the solvability of the optimization problem. Furthermore, we establish several equivalent conditions for the solvability of convex optimization problems. As applications, the notion of ‘exceptional family of elements’ for convex optimization problems is applied to the constrained optimization problem and convex quadratic programming problem and some existence results for solutions of these problems are obtained.

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Authors and Affiliations

  1. Department of Mathematics, Guangxi Normal University, Guilin, Guangxi, 541004, P.R. China

    Ren-you Zhong, Huan-xia Lian & Jiang-hua Fan

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  1. Ren-you Zhong
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  2. Huan-xia Lian
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  3. Jiang-hua Fan
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Correspondence to Jiang-hua Fan.

Additional information

This work was supported by the National Natural Science Foundation of China (11061006), the National Natural Science Foundation of China (11226224), the Program for Excellent Talents in Guangxi Higher Education Institutions, the Guangxi Natural Science Foundation (2012GXNSFBA053008) and the Initial Scientific Research Foundation for PHD of Guangxi Normal University.

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Zhong, Ry., Lian, Hx. & Fan, Jh. Exceptional Families of Elements for Optimization Problems in Reflexive Banach Spaces with Applications. J Optim Theory Appl 159, 341–359 (2013). https://doi.org/10.1007/s10957-013-0342-4

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  • DOI: https://doi.org/10.1007/s10957-013-0342-4

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