Constrained Extremum Problems and Image Space Analysis–Part I: Optimality Conditions

Image space analysis is a new tool for studying scalar and vector constrained extremum problems as well as generalized systems. In the last decades, the introduction of image space analysis has shown that the image space associated with the given problem provides a natural environment for the Lagran...

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Veröffentlicht in:	Journal of optimization theory and applications Jg. 177; H. 3; S. 609 - 636
Hauptverfasser:	Li, Shengjie, Xu, Yangdong, You, Manxue, Zhu, Shengkun
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.06.2018 Springer Nature B.V
Schlagworte:	Applications of Mathematics Calculus of Variations and Optimal Control; Optimization Engineering Inequalities Mathematical analysis Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Theory of Computation Optimality condition 90C29 Constrained extremum problem Image space analysis 49K05 Regularity condition
ISSN:	0022-3239, 1573-2878
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Abstract	Image space analysis is a new tool for studying scalar and vector constrained extremum problems as well as generalized systems. In the last decades, the introduction of image space analysis has shown that the image space associated with the given problem provides a natural environment for the Lagrange theory of multipliers and that separation arguments turn out to be a fundamental mathematical tool for explaining, developing and improving such a theory. This work, with its 3 parts, aims at contributing to describe the state-of-the-art of image space analysis for constrained optimization and to stress that it allows us to unify and generalize the several topics of optimization. In this 1st part, after a short introduction of such an analysis, necessary and sufficient optimality conditions are treated. Duality and penalization are the contents of the 2nd part. The 3rd part deals with generalized systems, in particular, variational inequalities and Ky Fan inequalities. Some further developments are discussed in all the parts.
AbstractList	Image space analysis is a new tool for studying scalar and vector constrained extremum problems as well as generalized systems. In the last decades, the introduction of image space analysis has shown that the image space associated with the given problem provides a natural environment for the Lagrange theory of multipliers and that separation arguments turn out to be a fundamental mathematical tool for explaining, developing and improving such a theory. This work, with its 3 parts, aims at contributing to describe the state-of-the-art of image space analysis for constrained optimization and to stress that it allows us to unify and generalize the several topics of optimization. In this 1st part, after a short introduction of such an analysis, necessary and sufficient optimality conditions are treated. Duality and penalization are the contents of the 2nd part. The 3rd part deals with generalized systems, in particular, variational inequalities and Ky Fan inequalities. Some further developments are discussed in all the parts.
Author	You, Manxue Zhu, Shengkun Xu, Yangdong Li, Shengjie
Author_xml	– sequence: 1 givenname: Shengjie surname: Li fullname: Li, Shengjie email: lisj@cqu.edu.cn organization: College of Mathematics and Statistics, Chongqing University – sequence: 2 givenname: Yangdong surname: Xu fullname: Xu, Yangdong organization: College of Science, Chongqing University of Posts and Telecommunications – sequence: 3 givenname: Manxue surname: You fullname: You, Manxue organization: College of Mathematics and Statistics, Chongqing University – sequence: 4 givenname: Shengkun surname: Zhu fullname: Zhu, Shengkun organization: Department of Economics and Mathematics, Southwestern University of Finance and Economics
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