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Enclosure Methods for Systems of Polynomial Equations and Inequalities

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Enclosure Methods for Systems of Polynomial Equations and Inequalities Enclosure Methods for Systems of Polynomial Equations and Inequalities Dissertation zur Erlangung des akademischen Grades des Doktors der Naturwissenschaften (Dr. rer. nat.) an der Mathematisch-Naturwissenschaftliche Sektion Fachbereich Mathematik und Statistik vorgelegt von Andrew Paul Smith Tag der mündlichen Prüfung: 5. Oktober 2012 1. Referent: Prof. Dr. Jürgen Garloff, Universität Konstanz und HTWG Konstanz 2. Referent: Prof. Dr. Jon Rokne, University of Calgary Abstract Many problems in applied mathematics can be formulated as a system of nonlinear equa- tions or inequalities, and a broad subset are those problems consisting either partially or completely of multivariate polynomials. Broadly speaking, an ‘enclosure’ method attempts n to solve such a problem over a specified region of interest, commonly a box subset of R , where n is the number of unknowns. Subdivision (or branch-and-bound) is a commonly- applied scheme, wherein a starting box is successively subdivided into sub-boxes; sub-boxes for which a proof of non-existence of a solution can be completed are discarded. As a component of such, a boundary method attempts to exploit the properties of component functions over the boundary of a sub-box, without considering their behaviour within it. Two main types of non-existence proof are considered for polynomial systems over boxes or sub-boxes. Firstly, the topological degree of a system of equations (considered as a n n mapping from R to R ) can be computed over a sub-box, which possesses a root-counting property. Together with an enclosure for the determinant of the Jacobian, the existence (or otherwise) of roots in the sub-box can be ascertained. Secondly, and alternatively, a range-enclosing method can be used to compute a guaranteed outer enclosure for the range of a multivariate polynomial over a sub-box; if it does not contain zero, a root is excluded. The Bernstein expansion is used, due to the tightness of the enclosure yielded and its rate of convergence to the true enclosure. In both cases, interval arithmetic is employed to obtain guaranteed enclosures and ensure the rigour of these existence tests. Advances have been made in four main areas. Firstly, an existing recursive algorithm for the computation of topological degree is investigated in detail, including a complexity analysis, and algorithmic improvements are proposed. Secondly, a simple branch-and-bound method for systems of polynomial equations, utilising Bernstein expansion and an existence test by Miranda, is developed, and alternative subdivision strategies are considered. Thirdly, a major improvement of the Bernstein expansion itself is achieved with the development of the implicit Bernstein form, which exhibits greatly improved performance for many cate- gories of polynomials. Finally, several new methods are developed and compared for the computation of affine lower bounding functions for polynomials, which may be employed in branch-and-bound schemes for problems involving inequalities, such as global optimisation problems. Numerical results and an overview of the developed software are given. i Zusammenfassung Viele Probleme der angewandten Mathematik können als Systeme nichtlinearer Gleichungen oder Ungleichungen formuliert werden. Eine große Teilmenge davon besteht teilweise oder vollständig aus multivariablen Polynomen. Die Idee einer Einschließungsmethode löst ein solches Problem innerhalb einer relevanten Region. Gewöhnlich ist diese Region eine Box- n Teilmenge von R , wobei n die Anzahl der Unbekannten ist. Subdivision (oder Branch-and- Bound) ist ein oft verwendetes Schema, bei dem eine Anfangsbox sukzessiv in Subboxen unterteilt wird. Subboxen, für die die Nichtexistenz einer Lösung bewiesen werden kann, werden ausgeschlossen. Als Bestandteil dieses Schemas ist die Randmethode, die versucht die Randeigenschaften der Komponentenfunktionen bezüglich der Subbox auszunutzen ohne dabei das Verhalten innerhalb der Subbox zu betrachten. Die Nichtexistenzbeweise für Systeme polynomer Gleichungen über Boxen oder Subboxen werden durch zwei Haupttypen beschrieben. Der erste Typ kann durch den topologis- n n chen Grad eines Gleichungssystems (als eine Abbildung von R auf R ) über eine Sub- box beschrieben werden. Diese besitzt die Eigenschaft, die Anzahl der Wurzeln zählen zu können. Dabei kann die Existenz (oder Nichtexistenz) von Wurzeln in dieser Sub- box durch die Bestimmung einer Einschließung der Determinanten der Jacobi-Matrix er- mittelt werden. Als zweiter Grundtyp bildet die Bereichseinschließungmethode, die eine äußere Einschränkung des Bildbereichs eines multivariablen Polynoms durch eine Sub- box garantiert. Enthält dieses Intervall nicht Null, so kann auch das Vorkommen einer Wurzel ausgeschlossen werden. Fokus der Bernstein-Erweiterung ist die Bestimmung präzis- erer Grenzen für die Einschließung und eine schnellere Konvergenz gegen die exakte Ein- schließung. Um eine garantierte Einschließung und die Korrektheit dieses Existenz-Tests sicherzustellen, wird in beiden Fällen die Intervallarithmetik angewandt. Es wurden Fortschritte in vier Bereichen erzielt. Erstens wurde ein existierender rekur- siver Algorithmus für die Berechnung des topologischen Grads detailiert untersucht. Dabei liegt die Betonung auf die Komplexitätsanalyse und algorithmische Verbesserung und Er- weiterungen. Der zweite Bereich beschäftigt sich zunächst mit einer einfachen Branch-and- Bound-Methode für polynome Gleichungssysteme. Dafür wurden die Bernstein-Erweiterung und der Existenz-Test von Miranda verwendet. Weiterhin wurden alternative Strategien zur Subdivision in Betracht gezogen. Drittens wurde eine wesentliche Verbesserung der Bernstein-Erweiterung durch die Entwicklung der impliziten Bernstein-Form erzielt. Es zeigt sich eine starke Verbesserung der Leistung für viele Arten von Polynomen. Ab- schließend wurden mehrere neue Verfahren für die Berechnung von unteren affinen Gren- zfunktionen für Polynome entwickelt und miteinander verglichen. Diese neue Verfahren können für Branch-and-Bound-Verfahren bei Problemen mit Ungleichungen, wie z.B. glob- ale Optimierungsprobleme, eingesetzt werden. Numerische Ergebnisse und eine Übersicht der entwickelten Software liegen vor. ii Acknowledgements First and foremost, I would like to warmly thank the two supervisors of this work, Dan Richardson (during my time in Bath, U.K.) and Jürgen Garloff (during my time in Konstanz, Germany), for their expert, friendly, and productive supervision and co-operation, as well as their patience and generosity. I would like to thank a number of people with whom I have been fortunate enough to co-write research papers, for their productive co-operation, principally Jürgen Garloff, but also (in chronological order) Christian Jansson, Ismail Idriss, Laurent Granvilliers, Evgenija Popova, Horst Werkle, and Riad Al-Kasasbeh. I would like to thank Hermann Schichl and Evgenija Popova for making their software available to me and for their technical assistance. Thanks also to the authors and main- tainers of the filib++ software package, which I have used extensively. I would like to thank my family for their love and support; in particular my mother, who never gave up gently pestering me to complete this thesis during the excessively long hiatus — I can’t believe it’s finished it, either! I’d like to pay tribute to a number of people who have helped to keep me (somewhat) sane during this odyssey in various ways: From the gaming group, I’d like to mention Christoph, Volker, Jan, Ole, Martin, Stefan, Sünje, Regina, Kris, Tamara, Patrick, and Christine. From the Institute for Optical Systems (IOS) at the HTWG Konstanz, Dang Le, Alex, David, Marcus, Thomas, Jan, Andy, Manuel, Klaus, Matthias Franz, as well as several others. I’d like to thank Dang for his support and motivation during our mutual struggles to write up our theses! I’d also like to thank several friends in the U.K., Mark, Richard, Phil, Caroline and Stuart, and Russ for staying in contact (in spite of my frequent silences!) during my long sojourn in Germany. I would like to thank a number of people who have helped to arrange financial sup- port and employment in various ways: at the University of Bath, Dan Richardson; at the HTWG Konstanz, Jürgen Garloff, Horst Werkle, Florin Ionescu, and Riad Al-Kasasbeh (Al- Balqa Applied University, Jordan); at the University of Konstanz, Jürgen Garloff, Markus Schweighofer, Stefan Volkwein, and Rainer Janßen. In particular I would like to thank Riad for his motivational support and his helpful urging during the final stages of this work, as well as his friendship and his help with potential future employment. I would like to gratefully acknowledge several funding agencies that have supported this work, either directly or indirectly. The University of Bath and the Engineering and Physical Sciences Research Council (EPSRC), in the U.K., and the University of Konstanz, in Ger- many, have partially funded my doctoral research with stipendia. The German Research Foundation (DFG), the German Ministry of Education, Science, Research, and Technol- ogy, the State of Baden-Württemberg, and the University of Applied Sciences (HTWG) Konstanz, in Germany, have funded several research projects on which have I have worked together with Jürgen Garloff; some of that research appears in this thesis. I would like to thank the two referees, Jürgen Garloff and Jon Rokne, as well as the oral examiners, for taking the time to review my work. For proofreading, thanks to Volker Bürkel, Martin Spitzbarth, and Dang Le. iii
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