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dc.contributor.author
Aguilera, Néstor Edgardo
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dc.contributor.author
Morin, Pedro
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dc.date.available
2019-09-23T17:34:25Z
dc.date.issued
2008-11
dc.identifier.citation
Aguilera, Néstor Edgardo; Morin, Pedro; Approximating optimization problems over convex functions; Springer; Numerische Mathematik; 111; 1; 11-2008; 1-34
dc.identifier.issn
0029-599X
dc.identifier.uri
http://hdl.handle.net/11336/84142
dc.description.abstract
Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(Omega)$, and some problems in economics. In the continuous setting and assuming smoothness, the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in making discrete approximations two difficulties arise: the continuous solutions may be not smooth, and functions with positive semidefinite discrete Hessian need not be convex in a discrete sense. Previous work has concentrated on non-local descriptions of convexity, making the number of constraints to grow super-linearly with the number of nodes even in dimension 2, and these descriptions are very difficult to extend to higher dimensions. In this paper we propose a finite difference approximation using positive semidefinite programs and discrete Hessians, and prove convergence under very general conditions, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes. Using positive semidefinite programming codes, we show concrete examples of approximations to problems in two and three dimensions.
dc.format
application/pdf
dc.language.iso
eng
dc.publisher
Springer
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dc.rights
info:eu-repo/semantics/openAccess
dc.rights.uri
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.subject
Convex Functions
dc.subject
Optimization
dc.subject
Finite Differences
dc.subject.classification
Matemática Aplicada
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dc.subject.classification
Matemáticas
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dc.subject.classification
CIENCIAS NATURALES Y EXACTAS
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dc.title
Approximating optimization problems over convex functions
dc.type
info:eu-repo/semantics/article
dc.type
info:ar-repo/semantics/artículo
dc.type
info:eu-repo/semantics/publishedVersion
dc.date.updated
2019-09-20T14:17:02Z
dc.identifier.eissn
0945-3245
dc.journal.volume
111
dc.journal.number
1
dc.journal.pagination
1-34
dc.journal.pais
Alemania
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dc.description.fil
Fil: Aguilera, Néstor Edgardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina
dc.description.fil
Fil: Morin, Pedro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; Argentina
dc.journal.title
Numerische Mathematik
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dc.relation.alternativeid
info:eu-repo/semantics/altIdentifier/doi/https://doi.org/10.1007/s00211-008-0176-4
dc.relation.alternativeid
info:eu-repo/semantics/altIdentifier/url/https://link.springer.com/article/10.1007%2Fs00211-008-0176-4
dc.relation.alternativeid
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/0804.1693
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