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Euclidean Jordan Algebras and Variational Problems Under Conic Constraints David Sossa

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Euclidean Jordan Algebras and Variational Problems Under Conic Constraints David Sossa Euclidean Jordan algebras and variational problems under conic constraints David Sossa To cite this version: David Sossa. Euclidean Jordan algebras and variational problems under conic constraints. Com- mutative Algebra [math.AC]. Université d’Avignon; Universidad de Chile, 2014. English. NNT : 2014AVIG0412. tel-01231470 HAL Id: tel-01231470 https://tel.archives-ouvertes.fr/tel-01231470 Submitted on 20 Nov 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ACADÉMIE D’AIX-MARSEILLE UNIVERSITÉ D’AVIGNON ET DES PAYS DE VAUCLUSE THÉSE DE DOCTORAT présentée á l’Université d’Avignon et des Pays de Vaucluse pour l’obtention du grade de Docteur SPECIALITE : Mathématiques Algèbres de Jordan euclidiennes et problèmes variationels avec contraintes coniques par David SOSSA soutenue le 4 septembre 2014 devant un jury composé de A. DANIILIDIS Université du Chili Rapporteur P. GAJARDO Université Federico Santa María Examinateur S. GOWDA Université du Maryland Rapporteur D.T. LUC Université d’Avignon Examinateur H. RAMÍREZ Université du Chili Directeur A. SEEGER Université d’Avignon Directeur M. VOLLE Université d’Avignon Examinateur Résumé Cette thèse concerne quatre thèmes apparemment différents, mais en fait étroitement liés: des problèmes variationnels sur les algèbres de Jordan euclidiennes, des problèmes de complé- mentarité sur l’espace des matrices symétriques, l’analyse angulaire entre deux cônes convexes fermés et analyse du chemin central en programmation conique symétrique. Dans la première partie de ce travail, le concept de “commutation au sens opérationnel” dans les algèbres de Jordan euclidiennes est étudié en fournissant un principe de commutation pour les problèmes variationnels avec des données spectrales. Dans la deuxième partie, nous abordons l’analyse et la résolution numérique d’une large classe de problèmes de complémentarité sur l’espace des matrices symétriques. Les conditions de complémentarité sont exprimées en termes de l’ordre de Loewner ou, plus généralement, en termes d’un cône du type Loewnerien. La troisième partie de ce travail est une tentative de construction d’une théorie générale des angles critiques pour une paire de cônes convexes fermés. L’analyse angulaire pour une paire de cônes spécialement structurés est également considérée. Par-exemple, nous travaillons avec des sous-espaces linéaires, des cônes polyédriques, des cônes de révolution, des cônes “topheavy” et des cônes de matrices. La dernière partie de ce travail étudie la convergence et le comportement asymptotique du chemin central en programmation conique symétrique. Ceci est fait en utilisant des techniques propres aux algèbres de Jordan. iii Abstract This thesis deals with four different but interrelated topics: variational problems on Euclidean Jordan algebras, complementarity problems on the space of symmetric matrices, angular analysis between two closed convex cones and the central path for symmetric cone linear programming. In the first part of this work we study the concept of “operator commutation” in Euclidean Jordan algebras by providing a commutation principle for variational problems involving spectral data. Our main concern of the second part is the analysis and numerical resolution of a broad class of complementarity problems on spaces of symmetric matrices. The complementarity conditions are expressed in terms of the Loewner ordering or, more generally, with respect to a dual pair of Loewnerian cones. The third part of this work is an attempt to build a general theory of critical angles for a pair of closed convex cones. The angular analysis for a pair of specially structured cones is also covered. For instance, we work with linear subspaces, polyhedral cones, revolution cones, topheavy cones and cones of matrices. The last part of this work focuses on the convergence and the limiting behavior of the central path in symmetric cone linear programming. This is done by using Jordan-algebra techniques. v Resumen En esta tesis doctoral se abordan cuatro tópicos diferentes pero mutuamente relacionados: Problemas variacionales sobre álgebras de Jordan Euclideanos, problemas de complemen- tariedad sobre espacios de matrices simétricas, análisis angular entre dos conos convexos y cerrados, y el camino central en programación cónica simétrica. La primera parte de este trabajo corresponde al estudio del concepto de “operator com- mutation” en álgebras de Jordan Euclideanos por medio del establecimiento de un principio de conmutación para problemas variacionales los cuales poseen datos espectrales. El principal enfoque de la segunda parte es el análisis y resolución numérica de una amplia clase de problemas de complementariedad formuladas en espacios de matrices simétricas. Las condiciones de complementariedad son expresadas en términos de la ordenación de Loewner o, mas general, con respecto a un par dual de conos Loewnerianos. En la tercera parte presentamos una construcción de la teoría general de ángulos críticos para pares de conos convexos y cerrados. El análisis angular de pares de conos con estructuras especiales es también abordada. Por ejemplo, en nuestro estudio incluimos: subespacios lineales, conos poliedrales, conos de revolución, conos topheavy y conos de matrices. La última parte de este trabajo está dedicada al estudio de la convergencia del camino central y del comportamiento de su punto límite en programación cónica simétrica. Esto es hecho por medio del uso de herramientas de álgebras de Jordan. vii ❆ ♠✐ ♣❛❞❡ ❨♦❧❛ ② ❱✐❝♦✐❛♥♦✱ ❛ ♠✐ ❡♣♦❛ ▲✐③ ❋❛❜✐♦❧❛ ② ❛ ♠✐ ❤✐❥❛ ❋❧♦ ❉❛❧✐③✳ ❈♦♥ ♠✉❝❤♦ ❆♠♦✳ Acknowledgments I thank God for blessing me much more than I deserve and for his wonderful promise expressed in John 3:16 “For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life.” I would like to thank my two advisors, Prof. Héctor Ramírez and Prof. Alberto Seeger. Their patience, encouragement, and immense knowledge were key motivations throughout my PhD. It is a real honor for me to have worked with them. Special thanks for the friendship and the hospitality received during my stay in Santiago and Avignon. I would like to thank Prof. Aris Daniilidis and Prof. Seetharama Gowda for review the earlier version of this thesis and for several suggestions that improved the presentation. I am also grateful to my oral committee members: Prof. Pedro Gajardo, Prof. Dinh The Luc and Prof. Michel Volle for their time, interest, helpful comments and insightful questions. I would also like to express my gratitude to all my friends I made during this period of time at the DIM-UCH and at the LMA-UAPV. Special thanks to the couples Paola and César, Ying and Huyuan. Thank you for all of your support and sincere friendship. Special thanks to Caroline for her help and warm hospitality offered during my stay in Avignon. I gratefully acknowledge the funding sources that made my Ph.D. work possible. My studies in Chile were funded by CONICYT (Comisión Nacional de Investigación Científica y Tecnológica de Chile) and during my stay in France I was funded by the grants for short research stays from the Universidad de Chile, from the Center for Mathematical Modeling (CMM) and from the Embassy of France in Chile. Last but not least, I thank to my family. Thanks to my beloved wife Liz Fabiola and my beloved daughter Flor Daliz. Thank you for being there every step of the way with me. You were the key motivation for this achievement. Thanks for filling my life with happiness. Thanks to my dear parents Yola and Victoriano for their support, prayers and love. Thanks to my sisters Ana, Isabel, Marcela, Verónica and my brothers René and Efraín for be with me in all time. xi Contents Résumé iii Abstract v Resumen vii Acknowledgments xi Introduction 1 1 Commutation principle for variational problems on EJA 9 1.1 Introduction .................................... 9 1.2 Preliminary material on Euclidean Jordan algebras .............. 11 1.2.1 The tangent space to the set of ordered Jordan frames ........ 12 1.3 Proof of the general commutation principle ................... 16 1.4 Applications .................................... 18 1.4.1 Variational inequalities .......................... 19 1.4.2 Distance to a spectral set ........................ 19 1.4.3 Inradius of a spectral cone ........................ 21 Bibliography ...................................... 22 2 Complementarity problems with respect to Loewnerian cones 25 2.1 Introduction .................................... 25 xiii Contents 2.2 Complementarity function approach for solving the SDCP .......... 27 2.2.1 Numerical experiments ......................... 28 2.2.2 A brief comment on the squaring technique ............... 33 2.3 On Loewnerian cones ............................... 34 2.3.1 Complementarity relative to Loewnerian cones ............. 36 2.3.2 Examples of Loewnerian cones and counter-examples ......... 37 2.4 By way of application: Finding the nearest Euclidean distance matrix ... 41 2.5 By way of conclusion ..............................
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