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

Kaiwen Meng, Xiaoqi Yang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

3 Citations (Scopus)

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 $$\frac{2}{3}$$23, 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.
Original language English 720-752 33 Journal of Optimization Theory and Applications 165 3 https://doi.org/10.1007/s10957-014-0664-x Published - 1 Jun 2015

Keywords

• KKT condition
• Nonlinear programming problem
• Regular subdifferential
• Second-order necessary condition
• Subderivative

ASJC Scopus subject areas

• Control and Optimization
• Management Science and Operations Research
• Applied Mathematics