Second‐Order Optimality Conditions for Scalar and Vector Optimization Problems in Banach Spaces

In this paper we present a very general and unified theory of second-order optimality conditions for general optimization problems subject to abstract constraints in Banach spaces. Our results apply both to the scalar case and the multicriteria case. Our approach rests essentially on the use of a si...

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Bibliographic Details
Published in:SIAM journal on control and optimization Vol. 45; no. 3; pp. 972 - 997
Main Author: Gfrerer, Helmut
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2006
Subjects:
Applied mathematics
Banach spaces
Computational mathematics
Inequality
Mathematical programming
Neighborhoods
Optimization
ISSN:0363-0129, 1095-7138
Online Access:Get full text
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Summary:In this paper we present a very general and unified theory of second-order optimality conditions for general optimization problems subject to abstract constraints in Banach spaces. Our results apply both to the scalar case and the multicriteria case. Our approach rests essentially on the use of a signed distance function for characterizing metric regularity of a certain multifunction associated with the problem. We prove variational results which show that, in a certain sense, our results are the best possible that one can obtain by using second-order analysis. We demonstrate how recently devised optimality conditions can be derived from our general framework, how they can be extended under weakened assumptions from the scalar case to the multiobjective case, and even how some new results also can be obtained.
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ISSN:0363-0129
1095-7138
DOI:10.1137/040612713