First- and Second-Order Necessary Conditions Via Exact Penalty Functions

Kaiwen Meng () and Xiaoqi Yang ()
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Kaiwen Meng: Southwest Jiaotong University
Xiaoqi Yang: The Hong Kong Polytechnic University

Journal of Optimization Theory and Applications, 2015, vol. 165, issue 3, No 3, 720-752

Abstract: Abstract In this paper, we study first- and second-order necessary conditions for nonlinear programming problems from the viewpoint of exact penalty functions. By applying the variational description of regular subgradients, we first establish necessary and sufficient conditions for a penalty term to be of KKT-type by using the regular subdifferential of the penalty term. In terms of the kernel of the subderivative of the penalty term, we also present sufficient conditions for a penalty term to be of KKT-type. We then derive a second-order necessary condition by assuming a second-order constraint qualification, which requires that the second-order linearized tangent set is included in the closed convex hull of the kernel of the parabolic subderivative of the penalty term. In particular, for a penalty term with order $$\frac{2}{3}$$ 2 3 , by assuming the nonpositiveness of a sum of a second-order derivative and a third-order derivative of the original data and applying a third-order Taylor expansion, we obtain the second-order necessary condition.

Keywords: Nonlinear programming problem; KKT condition; Second-order necessary condition; Subderivative; Regular subdifferential; 49J53; 49K99; 65K10 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10957-014-0664-x

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