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 |
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Pages (from-to) | 720-752 |
Number of pages | 33 |
Journal | Journal of Optimization Theory and Applications |
Volume | 165 |
Issue number | 3 |
DOIs | |
Publication status | 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