Second-Order Necessary Conditions for Nonlinear Optimization Problems with Abstract Constraints: The Degenerate Case

In this paper we derive second-order necessary conditions for optimality for an optimization problem with abstract constraints in Banach spaces. Results for the nondegenerate case derived earlier [H. Gfrerer, SIAM J. Control Optim., 45 (2006), pp. 972-997] are extended to the degenerate case. For th...

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Bibliographic Details
Published in:SIAM journal on optimization Vol. 18; no. 2; pp. 589 - 612
Main Author: Gfrerer, Helmut
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2007
Subjects:
Applied mathematics
Banach spaces
Computational mathematics
Inequality
Mathematical programming
Optimization
ISSN:1052-6234, 1095-7189
Online Access:Get full text
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Summary:In this paper we derive second-order necessary conditions for optimality for an optimization problem with abstract constraints in Banach spaces. Results for the nondegenerate case derived earlier [H. Gfrerer, SIAM J. Control Optim., 45 (2006), pp. 972-997] are extended to the degenerate case. For the mathematical programming problem, where the constraints are given by equality and finitely many inequality constraints, our approach applies to the degenerate case, when the equality constraints are not regular; our results appear to be new even in this special case. Our second-order necessary conditions are contained in the gap between the standard necessary and sufficient conditions, where the only difference is the change from a nonstrict to a strict inequality. Our results are formulated in such a way to be applicable also to vector optimization problems.
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ISSN:1052-6234
1095-7189
DOI:10.1137/050641387