DOCSLIB.ORG

Euclidean Jordan Algebras and Variational Problems Under Conic Constraints David Sossa

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

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 ..............................
Recommended publications
  • Arxiv:2009.05574V4 [Hep-Th] 9 Nov 2020 Predict a New Massless Spin One Boson [The ‘Lorentz’ Boson] Which Should Be Looked for in Experiments

    Arxiv:2009.05574V4 [Hep-Th] 9 Nov 2020 Predict a New Massless Spin One Boson [The ‘Lorentz’ Boson] Which Should Be Looked for in Experiments

    Trace dynamics and division algebras: towards quantum gravity and unification Tejinder P. Singh Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India e-mail: [email protected] Accepted for publication in Zeitschrift fur Naturforschung A on October 4, 2020 v4. Submitted to arXiv.org [hep-th] on November 9, 2020 ABSTRACT We have recently proposed a Lagrangian in trace dynamics at the Planck scale, for unification of gravitation, Yang-Mills fields, and fermions. Dynamical variables are described by odd- grade (fermionic) and even-grade (bosonic) Grassmann matrices. Evolution takes place in Connes time. At energies much lower than Planck scale, trace dynamics reduces to quantum field theory. In the present paper we explain that the correct understanding of spin requires us to formulate the theory in 8-D octonionic space. The automorphisms of the octonion algebra, which belong to the smallest exceptional Lie group G2, replace space- time diffeomorphisms and internal gauge transformations, bringing them under a common unified fold. Building on earlier work by other researchers on division algebras, we propose the Lorentz-weak unification at the Planck scale, the symmetry group being the stabiliser group of the quaternions inside the octonions. This is one of the two maximal sub-groups of G2, the other one being SU(3), the element preserver group of octonions. This latter group, coupled with U(1)em, describes the electro-colour symmetry, as shown earlier by Furey. We arXiv:2009.05574v4 [hep-th] 9 Nov 2020 predict a new massless spin one boson [the `Lorentz' boson] which should be looked for in experiments.
  • Part I. Origin of the Species Jordan Algebras Were Conceived and Grew to Maturity in the Landscape of Physics

    Part I. Origin of the Species Jordan Algebras Were Conceived and Grew to Maturity in the Landscape of Physics

    1 Part I. Origin of the Species Jordan algebras were conceived and grew to maturity in the landscape of physics. They were born in 1933 in a paper \Uber VerallgemeinerungsmÄoglichkeiten des Formalismus der Quantenmechanik" by the physicist Pascual Jordan; just one year later, with the help of John von Neumann and Eugene Wigner in the paper \On an algebraic generalization of the quantum mechanical formalism," they reached adulthood. Jordan algebras arose from the search for an \exceptional" setting for quantum mechanics. In the usual interpretation of quantum mechanics (the \Copenhagen model"), the physical observables are represented by Hermitian matrices (or operators on Hilbert space), those which are self-adjoint x¤ = x: The basic operations on matrices or operators are multiplication by a complex scalar ¸x, addition x + y, multipli- cation xy of matrices (composition of operators), and forming the complex conjugate transpose matrix (adjoint operator) x¤. This formalism is open to the objection that the operations are not \observable," not intrinsic to the physically meaningful part of the system: the scalar multiple ¸x is not again hermitian unless the scalar ¸ is real, the product xy is not observable unless x and y commute (or, as the physicists say, x and y are \simultaneously observable"), and the adjoint is invisible (it is the identity map on the observables, though nontrivial on matrices or operators in general). In 1932 the physicist Pascual Jordan proposed a program to discover a new algebraic setting for quantum mechanics, which would be freed from dependence on an invisible all-determining metaphysical matrix structure, yet would enjoy all the same algebraic bene¯ts as the highly successful Copenhagen model.
  • Geometry of the Shilov Boundary of a Bounded Symmetric Domain Jean-Louis Clerc

    Geometry of the Shilov Boundary of a Bounded Symmetric Domain Jean-Louis Clerc

    Geometry of the Shilov Boundary of a Bounded Symmetric Domain Jean-Louis Clerc To cite this version: Jean-Louis Clerc. Geometry of the Shilov Boundary of a Bounded Symmetric Domain. journal of geometry and symmetry in physics, 2009, Vol. 13, pp. 25-74. hal-00381665 HAL Id: hal-00381665 https://hal.archives-ouvertes.fr/hal-00381665 Submitted on 6 May 2009 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. Geometry of the Shilov Boundary of a Bounded Symmetric Domain Jean-Louis Clerc today Abstract In the first part, the theory of bounded symmetric domains is pre- sented along two main approaches : as special cases of Riemannian symmetric spaces of the noncompact type on one hand, as unit balls in positive Hermitian Jordan triple systems on the other hand. In the second part, an invariant for triples in the Shilov boundary of such a domain is constructed. It generalizes an invariant constructed by E. Cartan for the unit sphere in C2 and also the triple Maslov index on the Lagrangian manifold. 1 Introduction The present paper is an outgrowth of the cycle of conferences delivred by the author at the Tenth International Conference on Geometry, Integrability and Quantization, held in Varna in June 2008.
  • Spectral Cones in Euclidean Jordan Algebras

    Spectral Cones in Euclidean Jordan Algebras

    Spectral cones in Euclidean Jordan algebras Juyoung Jeong Department of Mathematics and Statistics University of Maryland, Baltimore County Baltimore, Maryland 21250, USA [email protected] and M. Seetharama Gowda Department of Mathematics and Statistics University of Maryland, Baltimore County Baltimore, Maryland 21250, USA [email protected] August 2, 2016 Abstract A spectral cone in a Euclidean Jordan algebra V of rank n is of the form K = λ−1(Q); where Q is a permutation invariant convex cone in Rn and λ : V!Rn is the eigenvalue map (which takes x to λ(x), the vector of eigenvalues of x with entries written in the decreasing order). In this paper, we describe some properties of spectral cones. We show, for example, that spectral cones are invariant under automorphisms of V, that the dual of a spectral cone is a spectral cone when V is simple or carries the canonical inner product, and characterize the pointedness/solidness of a spectral cone. We also show that for any spectral cone K in V, dim(K) 2 f0; 1; m − 1; mg, where dim(K) denotes the dimension of K and m is the dimension of V. Key Words: Euclidean Jordan algebra, spectral cone, automorphism, dual cone, dimension AMS Subject Classification: 15A18, 15A51, 15A57, 17C20, 17C30, 52A20 1 1 Introduction Let V be a Euclidean Jordan algebra of rank n and λ : V!Rn denote the eigenvalue map (which takes x to λ(x), the vector of eigenvalues of x with entries written in the decreasing order). A set E in V is said to be a spectral set [1] if there exists a permutation invariant set Q in Rn such that E = λ−1(Q): A function F : V!R is said to be a spectral function [1] if there is a permutation invariant function f : Rn !R such that F = f ◦ λ.
  • Arxiv:1106.4415V1 [Math.DG] 22 Jun 2011 R,Rno Udai Form

    Arxiv:1106.4415V1 [Math.DG] 22 Jun 2011 R,Rno Udai Form

    JORDAN STRUCTURES IN MATHEMATICS AND PHYSICS Radu IORDANESCU˘ 1 Institute of Mathematics of the Romanian Academy P.O.Box 1-764 014700 Bucharest, Romania E-mail: [email protected] FOREWORD The aim of this paper is to offer an overview of the most important applications of Jordan structures inside mathematics and also to physics, up- dated references being included. For a more detailed treatment of this topic see - especially - the recent book Iord˘anescu [364w], where sugestions for further developments are given through many open problems, comments and remarks pointed out throughout the text. Nowadays, mathematics becomes more and more nonassociative (see 1 § below), and my prediction is that in few years nonassociativity will govern mathematics and applied sciences. MSC 2010: 16T25, 17B60, 17C40, 17C50, 17C65, 17C90, 17D92, 35Q51, 35Q53, 44A12, 51A35, 51C05, 53C35, 81T05, 81T30, 92D10. Keywords: Jordan algebra, Jordan triple system, Jordan pair, JB-, ∗ ∗ ∗ arXiv:1106.4415v1 [math.DG] 22 Jun 2011 JB -, JBW-, JBW -, JH -algebra, Ricatti equation, Riemann space, symmet- ric space, R-space, octonion plane, projective plane, Barbilian space, Tzitzeica equation, quantum group, B¨acklund-Darboux transformation, Hopf algebra, Yang-Baxter equation, KP equation, Sato Grassmann manifold, genetic alge- bra, random quadratic form. 1The author was partially supported from the contract PN-II-ID-PCE 1188 517/2009. 2 CONTENTS 1. Jordan structures ................................. ....................2 § 2. Algebraic varieties (or manifolds) defined by Jordan pairs ............11 § 3. Jordan structures in analysis ....................... ..................19 § 4. Jordan structures in differential geometry . ...............39 § 5. Jordan algebras in ring geometries . ................59 § 6. Jordan algebras in mathematical biology and mathematical statistics .66 § 7.
  • Structure and Representations of Noncommutative Jordan Algebras

    Structure and Representations of Noncommutative Jordan Algebras

    STRUCTURE AND REPRESENTATIONS OF NONCOMMUTATIVE JORDAN ALGEBRAS BY KEVIN McCRIMMON(i) 1. Introduction. The first three sections of this paper are devoted to proving an analogue of N. Jacobson's Coordinatization Theorem [6], [7] for noncom- mutative Jordan algebras with n ^ 3 "connected" idempotents which charac- terizes such algebras as commutative Jordan algebras £)CD„,y) or split quasi- associative algebras £>„(;i).The proof proceeds by using certain deformations of the algebra and the base field which reduce the theorem to the special cases when the "indicator" 0 is ^ or 0, and then using the Peirce decomposition to show that in these cases the algebra is commutative or associative respectively. Next we develop the standard structure theory, using the Coordinatization Theorem to classify the simple algebras. The usual approach of A. A. Albert [2], [3] and R. H. Oehmke [9] proceeds by constructing an admissable trace function on the algebra 31 to show that ÎI + is a simple Jordan algebra, and then characterizing the possible multiplications in 31 yielding a given such 3I+. Our approach is in some sense more intrinsic, and has the advantage of dealing simul- taneously with all characteristics different from 2 ; the results for characteristic 3 seem to be new. Another advantage of using the Coordinatization Theorem is that it furnishes a representation theory (in the sense of Eilenberg [4]) for the class of noncom- mutative Jordan algebras. The usual theorems hold only for birepresentations involving algebras of degree _ 3 ; counterexamples are given to show they fail spectacularly for algebras of degree 1 or 2. In this paper 2t will always denote a noncommutative Jordan algebra (not necessarily finite-dimensional) over a field 3> of characteristic ^ 2.
  • Pre-Jordan Algebras

    Pre-Jordan Algebras

    MATH. SCAND. 112 (2013), 19–48 PRE-JORDAN ALGEBRAS DONGPING HOU, XIANG NI and CHENGMING BAI∗ Abstract The purpose of this paper is to introduce and study a notion of pre-Jordan algebra. Pre-Jordan algebras are regarded as the underlying algebraic structures of the Jordan algebras with a nondegen- erate symplectic form. They are the algebraic structures behind the Jordan Yang-Baxter equation and Rota-Baxter operators in terms of O-operators of Jordan algebras introduced in this paper. Pre-Jordan algebras are analogues for Jordan algebras of pre-Lie algebras and fit into a bigger framework with a close relationship with dendriform algebras. The anticommutator of a pre- Jordan algebra is a Jordan algebra and the left multiplication operators give a representation of the Jordan algebra, which is the beauty of such a structure. Furthermore, we introduce a notion of O-operator of a pre-Jordan algebra which gives an analogue of the classicalYang-Baxter equation in a pre-Jordan algebra. 1. Introduction 1.1. Motivations Jordan algebras were first studied in the 1930s in the context of axiomatic quantum mechanics ([6]) and appeared in many areas of mathematics like differential geometry ([30], [19], [35], [37], [44]), Lie theory ([31], [34]) and analysis ([37], [48]). A Jordan algebra can be regarded as an “opposite” of a Lie algebra in the sense that the commutator of an associative algebra is a Lie algebra and the anticommutator of an associative algebra is a Jordan algebra, although not every Jordan algebra is isomorphic to the anticommutator of an associative algebra (such a Jordan algebra is called special, otherwise, it is called exceptional).
  • [Math.OC] 2 Apr 2001

    [Math.OC] 2 Apr 2001

    UNIVERSITY OF CAMBRIDGE Numerical Analysis Reports SELF–SCALED BARRIERS FOR IRREDUCIBLE SYMMETRIC CONES Raphael Hauser and Yongdo Lim arXiv:math/0104020v1 [math.OC] 2 Apr 2001 DAMTP 2001/NA04 April 2001 Department of Applied Mathematics and Theoretical Physics Silver Street Cambridge England CB3 9EW SELF–SCALED BARRIERS FOR IRREDUCIBLE SYMMETRIC CONES RAPHAEL A. HAUSER, YONGDO LIM April 2, 2001 Abstract. Self–scaled barrier functions are fundamental objects in the theory of interior–point methods for linear optimization over symmetric cones, of which linear and semidefinite program- ming are special cases. We are classifying all self–scaled barriers over irreducible symmetric cones and show that these functions are merely homothetic transformations of the universal barrier func- tion. Together with a decomposition theorem for self–scaled barriers this concludes the algebraic classification theory of these functions. After introducing the reader to the concepts relevant to the problem and tracing the history of the subject, we start by deriving our result from first principles in the important special case of semidefinite programming. We then generalise these arguments to irreducible symmetric cones by invoking results from the theory of Euclidean Jordan algebras. Key Words Semidefinite programming, self–scaled barrier functions, interior–point methods, symmetric cones, Euclidean Jordan algebras. AMS 1991 Subject Classification Primary 90C25, 52A41, 90C60. Secondary 90C05, 90C20. Contact Information and Credits Raphael Hauser, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, England. [email protected]. Research supported in part by the Norwegian Research Council through project No. 127582/410 “Synode II”, by the Engineering and Physical Sciences Research Council of the UK under grant No.
  • Arxiv:1207.3214V1 [Math.MG] 13 Jul 2012 Ftosubspaces Two of Esc E.Tepouto W Oe Sjs H Sa Produc Usual the Just Is Cones Two of Product Cone the a Sets

    Arxiv:1207.3214V1 [Math.MG] 13 Jul 2012 Ftosubspaces Two of Esc E.Tepouto W Oe Sjs H Sa Produc Usual the Just Is Cones Two of Product Cone the a Sets

    SYMMETRIC CONES, THE HILBERT AND THOMPSON METRICS BOSCHÉ AURÉLIEN Abstract. Symmetric cones can be endowed with at least two in- teresting non Riemannian metrics: the Hilbert and the Thompson metrics. It is trivial that the linear maps preserving the cone are isometries for those two metrics. Oddly enough those are not the only isometries in general. We give here a full description of the isometry groups for both the Hilbert and the Thompson metrics using essentially the theory of euclidean Jordan algebras. Those results were already proved for the symmetric cone of complexe positive hermitian matrices by L. Molnár in [7]. In this paper how- ever we do not make any assumption on the symmetric cone under scrutiny (it could be reducible and contain exceptional factors). 1. Preliminaries A cone is a subset C of some euclidean space Rn that is invariant by positive scalar exterior multiplication. A convex cone is a cone that is also a convex subset of Rn. A cone C is proper (resp. open) if its closure contains no complete line (resp. if it’s interior is not empty). In this paper we deal exclusively with open proper cones and C will always be such a set. The product of two cones is just the usual product of sets. A cone C ∈ Rn is reducible if Rn splits orthogonally as the sum of two subspaces A and B each containing a cone CA and CB such that C is the product of CA and CB , i.e. the set of all sums of elements of CA n and CB.
  • Control System Analysis on Symmetric Cones

    Control System Analysis on Symmetric Cones

    Submitted, 2015 Conference on Decision and Control (CDC) http://www.cds.caltech.edu/~murray/papers/pm15-cdc.html Control System Analysis on Symmetric Cones Ivan Papusha Richard M. Murray Abstract— Motivated by the desire to analyze high dimen- Autonomous systems for which A is a Metzler matrix are sional control systems without explicitly forming computation- called internally positive,becausetheirstatespacetrajec- ally expensive linear matrix inequality (LMI) constraints, we tories are guaranteed to remain in the nonnegative orthant seek to exploit special structure in the dynamics matrix. By Rn using Jordan algebraic techniques we show how to analyze + if they start in the nonnegative orthant. As we will see, continuous time linear dynamical systems whose dynamics are the Metzler structure is (in a certain sense) the only natural exponentially invariant with respect to a symmetric cone. This matrix structure for which a linear program may generically allows us to characterize the families of Lyapunov functions be composed to verify stability. However, the inclusion that suffice to verify the stability of such systems. We highlight, from a computational viewpoint, a class of systems for which LP ⊆ SOCP ⊆ SDP stability verification can be cast as a second order cone program (SOCP), and show how the same framework reduces to linear motivates us to determine if stability analysis can be cast, for programming (LP) when the system is internally positive, and to semidefinite programming (SDP) when the system has no example, as a second order cone program (SOCP) for some special structure. specific subclass of linear dynamics, in the same way that it can be cast as an LP for internally positive systems, or an I.
  • A Characterization of Symmetric Cones by an Order-Reversing Property of the Pseudoinverse Maps

    A Characterization of Symmetric Cones by an Order-Reversing Property of the Pseudoinverse Maps

    A CHARACTERIZATION OF SYMMETRIC CONES BY AN ORDER-REVERSING PROPERTY OF THE PSEUDOINVERSE MAPS CHIFUNE KAI Abstract. When a homogeneous convex cone is given, a natural partial order is introduced in the ambient vector space. We shall show that a homogeneous convex cone is a symmetric cone if and only if Vinberg’s ¤-map and its inverse reverse the order. Actually our theorem is formulated in terms of the family of pseudoinverse maps including the ¤-map, and states that the above order-reversing property is typical of the ¤-map of a symmetric cone which coincides with the inverse map of the Jordan algebra associated with the symmetric cone. 1. Introduction Let Ω be an open convex cone in a finite-dimensional real vector space V which is regular, that is, Ω\(¡Ω) = f0g, where Ω stands for the closure of Ω. For x; y 2 V , we write x ºΩ y if x ¡ y 2 Ω. Clearly, this defines a partial order in V . In the special case that Ω is the one-dimensional symmetric cone R>0, the order ºΩ is the usual one and is reversed by taking inverse numbers in R>0. In general, let a symmetric cone Ω ½ V be given. Then V has a structure of the Jordan algebra associated with Ω. In this case, the Jordan algebra inverse map is an involution on Ω and reverses the order ºΩ. This order-reversing property helps our geometric understanding of the Jordan algebra inverse map. In this paper, we shall investigate this order-reversing property in a more general setting and give a characterization of symmetric cones.
  • Spectral Functions and Smoothing Techniques on Jordan Algebras

    Spectral Functions and Smoothing Techniques on Jordan Algebras

    Universite¶ catholique de Louvain Faculte¶ des Sciences appliquees¶ Departement¶ d'Ingenierie¶ mathematique¶ Center for Operation Research and Econometrics Center for Systems Engineering and Applied Mechanics Spectral Functions and Smoothing Techniques on Jordan Algebras Michel Baes Thesis submitted in partial ful¯llment of the requirements for the degree of Docteur en Sciences Appliqu¶ees Dissertation Committee: Fran»coisGlineur Universit¶ecatholique de Louvain Yurii Nesterov (advisor) Universit¶ecatholique de Louvain Cornelius Roos Technische Universiteit Delft Jean-Pierre Tignol Universit¶ecatholique de Louvain Paul Van Dooren (advisor) Universit¶ecatholique de Louvain Jean-Philippe Vial Universit¶ede Gen`eve Vincent Wertz (chair) Universit¶ecatholique de Louvain ii Contents List of notation vii 1 Introduction and preliminaries 1 1.1 Comparing algorithms .............................. 2 1.2 Linear Programming ............................... 3 1.3 Convex Programming .............................. 5 1.4 Self-scaled Optimization, and formally real Jordan algebras ......... 10 1.5 A closer look at interior-point methods ..................... 12 1.5.1 Newton's Algorithm: solving unconstrained problems ........ 12 1.5.2 Barrier methods: dealing with constraints ............... 13 1.5.3 Choosing an appropriate barrier .................... 13 1.5.4 Path-following interior-point methods for Linear Programming ... 17 1.5.5 Path-following interior-point methods for Self-Scaled Programming . 18 1.6 Smoothing techniques .............................. 20 1.7 Eigenvalues in Jordan algebra make it work: more applications ....... 22 1.7.1 A concavity result ............................ 22 1.7.2 Augmented barriers in Jordan algebras ................. 23 1.8 Overview of the thesis and research summary ................. 25 2 Jordan algebras 27 2.1 The birth of Jordan algebras .......................... 28 2.2 Algebras and Jordan algebras .........................