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University of Groningen the Copositive Cone, the Completely University of Groningen The copositive cone, the completely positive cone and their generalisations Dickinson, Peter James Clair IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2013 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Dickinson, P. J. C. (2013). The copositive cone, the completely positive cone and their generalisations. s.n. Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: 27-09-2021 The Copositive Cone, the Completely Positive Cone and their Generalisations Peter J.C. Dickinson This PhD project was carried out at the Johann Bernoulli Institute according to the requirements of the Graduate School of Science (Faculty of Mathematics and Natural Sciences, University of Groningen). Funding was provided to me as a bursary student from the Rosalind Franklin Fellowship programme of the University of Groningen. Rijksuniversiteit Groningen The Copositive Cone, the Completely Positive Cone and their Generalisations Proefschrift ter verkrijging van het doctoraat in de Wiskunde en Natuurwetenschappen aan de Rijksuniversiteit Groningen op gezag van de Rector Magnificus, dr. E. Sterken, in het openbaar te verdedigen op vrijdag 12 april 2013 om 16.15 uur door Peter James Clair Dickinson geboren op 16 maart 1985 te Ealing, Engeland Promotor: Prof. dr. M. D¨ur Beoordelingscommissie: Prof. dr. I.M. Bomze Prof. dr. M. Laurent Prof. dr. A.N. Letchford ISBN: 978-90-367-6101-7 (printed version) ISBN: 978-90-367-6107-9 (electronic version) Contents Acknowledgements v Personal contributions vii I Introduction 1 1 Introduction 3 1.1 Copositive & Completely Positive Cones . .3 1.2 Geometry and Optimisation . .7 1.2.1 Geometry . .7 1.2.2 Mathematical Optimisation . 10 1.2.3 Duality . 12 1.2.4 Special types of conic optimisation . 17 1.3 Maximum Weight Clique Problem . 18 2 Set-semidefinite Optimisation 23 2.1 Set-semidefinite Cone . 23 2.2 Single Quadratic Constraint Problem . 24 2.3 Standard Quadratic Optimisation Problem . 26 2.4 Quadratic Binary Optimisation . 27 2.5 Considering a special case . 32 3 Complexity 35 3.1 Membership Problems . 35 3.2 Ellipsoid Method . 37 3.3 Copositivity and Completely Positivity . 41 4 Sparse C∗ detection and decomposition 49 4.1 Rank-One Decomposition . 51 4.2 Indices of Degree Zero or One . 53 4.3 Chains . 57 i CONTENTS 4.4 Matrices with circular graphs . 60 4.5 Reducing Chain Lengths . 64 4.6 Preprocessing . 68 4.7 Number of minimal decompositions . 69 II Geometry 73 5 Proper Cones 75 6 The Set of Zeros 77 7 Interior of C and C∗ 79 7.1 Introduction . 79 7.2 Original Proof . 81 7.3 New Proof . 82 7.4 Concluding Remarks . 83 8 Facial Structure 85 8.1 Geometry of General Proper Cones . 85 8.2 C2 and C∗2 .............................. 89 8.3 Extreme Rays of C and C∗ ..................... 90 8.4 Maximal Faces of the Copositive Cone . 95 8.5 Maximal Faces of the Completely Positive Cone . 98 8.6 Lower Bound on Dimension of Maximal Faces of C∗ ....... 100 III Approximations 103 9 Simplicial Partitioning 107 9.1 Introduction . 107 9.1.1 Approximation hierarchy . 108 9.1.2 Partitioning methods . 110 9.2 Counter-examples . 112 9.3 An Exhaustive Partitioning Scheme . 118 9.4 Reconsidering unrestricted free bisection . 120 9.5 The importance of picking a fixed lambda or rho . 121 9.6 Conclusion . 124 10 Moment Approximations 125 10.1 Introduction . 125 10.1.1 Notation . 125 10.1.2 Contribution . 126 ii Copositive Cone, Completely Positive Cone & Generalisations CONTENTS 10.2 Introduction to Moments . 126 10.3 Lasserre's hierarchies . 127 10.4 New hierarcies . 129 10.5 Examples . 131 10.6 Conclusion . 133 11 Sum-of-Squares 135 11.1 Introduction . 135 11.2 Parrilo's approximation hierarchy . 137 r 11.3 Scaling a matrix out of Kn ..................... 137 1 11.4 Scaling a matrix into K5 ...................... 138 11.5 The importance of scaling to binary diagonals . 140 11.6 Dual of the Parrilo approximations . 141 12 Cones of Polynomials 145 12.1 Positivstellens¨atze . 145 12.2 Reconsidering polynomial constraints . 147 12.3 Application to Optimisation . 147 12.4 Inner Approximation Hierarchy . 149 Summary 151 Samenvatting 153 Bibliography 155 Nomenclature 163 Index 167 Peter J.C. Dickinson iii CONTENTS iv Copositive Cone, Completely Positive Cone & Generalisations Acknowledgements I have found my time as a PhD student to be both enjoyable and profitable. However, this would not have been possible without the help that I received from a great many people, and I would like to take this opportunity to thank some of them. First and foremost, I would like to thank my supervisor, Prof. Dr. Mirjam D¨ur.When coming to Mirjam to ask for the PhD position, I had never studied optimisation before and had been away from academia for two years. However, she must have seen some potential in me and gave me a chance, something that I will be eternally grateful for. Naturally, I would like to thank her for teaching me the subject matter that I have needed to know, however she has also done much more than that. When I started to write up my work, I produced little more than a mess of equations, but Mirjam taught me to write in a more legible manner. She also taught me how academia works and provided lots of advice for my future career. Most importantly of all, she made me feel welcome, and made my time as a PhD student an enjoyable one. There are many people that I would like to thank at the University of Groningen and in particular at the Johann Bernoulli Institute. I would like to thank all of my colleagues, who are too numerous to include a list of. Our lunches always gave me a nice break during the day. One person that I would like to include extra thanks to is Luuk Gijben for our enjoyable collaborations together and for translating the summary of this thesis into Dutch. In addition to my colleagues in Groningen, I would also like to thank my colleagues from around the world. Without them, the conferences that I went to would not have been as informative or enjoyable as they were. I wish to especially thank my reading committee, Prof. Dr. Immanuel Bomze, Prof. Dr. Monique Laurent and Prof. Dr. Adam Letchford, for the time and care that they took in considering my thesis. I would further like to thank colleagues that I have collaborated with during my PhD. Many of these colleagues I also visited, during which they made me feel very welcome. These colleagues are: Prof. Dr. Kurt Anstreicher, Prof. Dr. Samuel Burer, Prof. Dr. Gabriele Eichfelder, Dr. Roland Hildebrand and Prof. Dr. Janez Povh. Organisations that I would like to thank for making this PhD possible v ACKNOWLEDGEMENTS are the University of Groningen, the Johann Bernoulli Institute, the Gronin- gen Graduate School of Science and the Landelijk Netwerk Mathematische Besliskunde. I am also thankful for the funding that I received as a bursary student from the Rosalind Franklin Fellowship programme of the University of Groningen. Although I am grateful to these organisation, most people who know me will know that I have something verging on a hatred for forms and other aspects of dealing with administration. I would thus like to thank those who have helped to reduce my administrative burden, especially Esmee Elshof and Ineke Schelhaas. In addition to this, I would also like to thank Peter Arendz for his regular assistance in all things connected to computers. There also many people from outside of academia who have provided help to me. I am grateful for the support shown to me throughout my life by my family, especially Keith Dickinson, Claire Dickinson and Katie Dickinson. I would also like to thank the family of my girlfriend, especially Erik Bosman, Geertje Bosman-Wichers and Marije Bosman, who provided invaluable support from the start of my life in the Netherlands. I would also like to thank my friends in both England and the Netherlands for helping me to relax. In the Netherlands, this was primarily through the scuba diving club G.B.D. Calamari, who showed me friendship as well as providing me a place to continue my hobby of scuba diving. One final very big thanks goes to my girlfriend, Renske Bosman. Without her support through both the good and the tough times of my PhD, I never would have managed. I hope that I can show the same support to her when she does a PhD. vi Copositive Cone, Completely Positive Cone & Generalisations Personal contributions I began my PhD with Prof. Dr. Mirjam D¨urat the University of Groningen in April 2009, and during my time as a PhD student I have produced (along with my co-authors) the following articles: Published articles: [Dic13] Peter J.C.
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