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Optimality Conditions for Disjunctive Optimization in Reflexive Banach Spaces

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Abstract

We study optimization problems with the constraints having disjunctive structures in reflexive Banach spaces. By the representations of contingent cones and Fréchet normal cones to finite unions of sets in general Banach spaces and using the special structures of convex generalized polyhedral sets, we calculate the Mordukhovich normal cones to finite unions of closed and convex sets that particularly include convex generalized polyhedral sets in reflexive Banach spaces. Furthermore, based on these calculations and the Guignard-type constraint qualifications, we derive new optimality conditions for disjunctive optimization problems. We also present specializations of these results to optimization problems with variational inequality constraints.

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Acknowledgments

The authors are grateful to the anonymous referees who have contributed to improve the quality of the paper. The research was partially supported by the National Natural Sciences Grant (No. 11071052) and the Scientific Innovation Project for Graduate of Heilongjiang Province (No. YJSCX2012-160HLJ).

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  1. School of Mathematical Sciences, Harbin Normal University, Harbin,  150025, China

    Wen Song & Qianqian Wang

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  1. Wen Song
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  2. Qianqian Wang
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Correspondence to Wen Song.

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Song, W., Wang, Q. Optimality Conditions for Disjunctive Optimization in Reflexive Banach Spaces. J Optim Theory Appl 164, 436–454 (2015). https://doi.org/10.1007/s10957-014-0571-1

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