Journal of Optimization Theory and Applications

, Volume 171, Issue 1, pp 45–69

Second-Order Conditions for Open-Cone Minimizers and Firm Minimizers in Set-Valued Optimization Subject to Mixed Constraints

Article

DOI: 10.1007/s10957-016-0995-x

Cite this article as:
Khanh, P.Q. & Tung, N.M. J Optim Theory Appl (2016) 171: 45. doi:10.1007/s10957-016-0995-x
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Abstract

We consider second-order optimality conditions for set-valued optimization problems subject to mixed constraints. Such optimization models are useful in a wide range of practical applications. By using several kinds of derivatives, we obtain second-order necessary conditions for local Q-minimizers and local firm minimizers with attention to the envelope-like effect. Under the second-order Abadie constraint qualification, we get stronger necessary conditions. When the second-order Kurcyusz–Robinson–Zowe constraint qualification is imposed, our multiplier rules are of the Karush–Kuhn–Tucker type. Sufficient conditions for firm minimizers are established without any convexity assumptions. As an application, we extend and improve some recent existing results for nonsmooth mathematical programming.

Keywords

Set-valued optimizationAbadie constraint qualificationKurcyusz–Robinson–Zowe constraint qualificationOpen-cone minimizerFirm minimizer

Mathematics Subject Classification

90C2949J5290C4690C48

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsInternational University, Vietnam National University Hochiminh CityHochiminh CityVietnam
  2. 2.Department of Mathematics and ComputingUniversity of Science, Vietnam National University Hochiminh CityHochiminh CityVietnam