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New Constraint Qualification and Conjugate Duality for Composed Convex Optimization Problems.

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Title: New Constraint Qualification and Conjugate Duality for Composed Convex Optimization Problems.
Authors: Boƫ, R. I.1, Grad, S. M.1, Wanka, G.1 gert.wanka@mathematik.tu-chemnitz.de
Source: Journal of Optimization Theory & Applications. Nov2007, Vol. 135 Issue 2, p241-255. 15p.
Subjects: Mathematical optimization, Convex functions, Real variables, Conjugate direction methods, Numerical solutions to equations, Mathematics, Mathematical analysis, Numerical analysis, Mathematical variables
Abstract: We present a new constraint qualification which guarantees strong duality between a cone-constrained convex optimization problem and its Fenchel-Lagrange dual. This result is applied to a convex optimization problem having, for a given nonempty convex cone K, as objective function a K-convex function postcomposed with a K-increasing convex function. For this so-called composed convex optimization problem, we present a strong duality assertion, too, under weaker conditions than the ones considered so far. As an application, we rediscover the formula of the conjugate of a postcomposition with a K-increasing convex function as valid under weaker conditions than usually used in the literature. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Optimization Theory & Applications is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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DbLabel: Engineering Source
An: 27710644
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  Data: <searchLink fieldCode="DE" term="%22Mathematical+optimization%22">Mathematical optimization</searchLink><br /><searchLink fieldCode="DE" term="%22Convex+functions%22">Convex functions</searchLink><br /><searchLink fieldCode="DE" term="%22Real+variables%22">Real variables</searchLink><br /><searchLink fieldCode="DE" term="%22Conjugate+direction+methods%22">Conjugate direction methods</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+solutions+to+equations%22">Numerical solutions to equations</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics%22">Mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+analysis%22">Mathematical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+variables%22">Mathematical variables</searchLink>
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  Data: We present a new constraint qualification which guarantees strong duality between a cone-constrained convex optimization problem and its Fenchel-Lagrange dual. This result is applied to a convex optimization problem having, for a given nonempty convex cone K, as objective function a K-convex function postcomposed with a K-increasing convex function. For this so-called composed convex optimization problem, we present a strong duality assertion, too, under weaker conditions than the ones considered so far. As an application, we rediscover the formula of the conjugate of a postcomposition with a K-increasing convex function as valid under weaker conditions than usually used in the literature. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Journal of Optimization Theory & Applications is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s10957-007-9247-4
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        Text: English
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        Type: general
      – SubjectFull: Convex functions
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      – SubjectFull: Real variables
        Type: general
      – SubjectFull: Conjugate direction methods
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      – SubjectFull: Numerical solutions to equations
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      – SubjectFull: Mathematical variables
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      – TitleFull: New Constraint Qualification and Conjugate Duality for Composed Convex Optimization Problems.
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              Text: Nov2007
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