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



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.


Set-valued optimization Abadie constraint qualification Kurcyusz–Robinson–Zowe constraint qualification Open-cone minimizer Firm minimizer 

Mathematics Subject Classification

90C29 49J52 90C46 90C48 


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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

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