Computing uniform convex approximations for convex envelopes and convex hulls

Rida Laraki (rida.laraki@dauphine.fr) and Jean-Bernard Lasserre (lasserre@laas.fr)
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Rida Laraki: CECO - Laboratoire d'économétrie de l'École polytechnique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique
Jean-Bernard Lasserre: LAAS-MAC - Équipe Méthodes et Algorithmes en Commande - LAAS - Laboratoire d'analyse et d'architecture des systèmes - UT Capitole - Université Toulouse Capitole - UT - Université de Toulouse - INSA Toulouse - Institut National des Sciences Appliquées - Toulouse - INSA - Institut National des Sciences Appliquées - UT - Université de Toulouse - UT2J - Université Toulouse - Jean Jaurès - UT - Université de Toulouse - UT3 - Université Toulouse III - Paul Sabatier - UT - Université de Toulouse - CNRS - Centre National de la Recherche Scientifique - Toulouse INP - Institut National Polytechnique (Toulouse) - UT - Université de Toulouse

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Abstract: We provide a numerical procedure to compute uniform (convex) approximations {f_{r}} of the convex envelope f of a rational fraction f, on a compact semi-algebraic set D. At each point x in K=co(D), computing f_{r}(x) reduces to solving a semidefinite program. We next characterize the convex hull K=co(D) in terms of the projection of a semi-infinite LMI, and provide outer convex approximations {K_{r}}?K. Testing whether x is not in K reduces to solving finitely many semidefinite programs.

Keywords: Convex envelope; Semi-definite program; Semi-algbraic set; Duality; Enveloppe convexe; Programme semi-défini; Ensemble semi-algébrique; Dualité (search for similar items in EconPapers)
Date: 2008
Note: View the original document on HAL open archive server: https://hal.science/hal-00243009v1
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Citations: View citations in EconPapers (5)

Published in Journal of Convex Analysis, 2008, 15 (3), pp.635-654

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