Spectrahedral and semidefinite representability of orbitopes Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften vorgelegt von Kobert, Tim an der Mathematisch-naturwissenschaftliche Sektion Fachbereich Mathematik & Statistik Konstanz, 2019 Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-2-15qzbd2l8d9fi3 Tag der m¨undlichen Pr¨ufung:14.12.2018 1. Referent: Prof. Dr. Claus Scheiderer 2. Referent: Prof. Dr. Daniel Plaumann 1 for Alexandra Contents 1 Introduction 6 1.1 Motivation . .6 1.2 Acknowledgments . .6 1.3 Related work . .6 1.4 Brief summary of main results . .9 1.5 Detailed overview . .9 2 Preliminaries 14 2.1 General notations . 14 2.2 Convex geometry . 15 2.3 Algebraic Geometry . 17 2.4 Lie algebras . 20 2.5 Semidefinite programming . 25 2.6 Orbitopes . 27 3 Polar orbitopes 30 3.1 Spectrahedron property . 31 3.2 Correspondence of face orbits . 39 3.3 The momentum polytope . 43 3.3.1 Restricted weights . 43 3.3.2 Polyhedral description of the momentum polytope . 47 3.4 Minimality conditions . 55 3.4.1 Spectrahedral representations of minimal size . 57 3.4.2 Equivariant SDP lifts . 62 3.5 Doubly spectrahedral polar orbitopes . 67 3.6 Example - Stiefel manifolds . 69 4 Toric orbitopes 76 4.1 The multivariate Carath´eodory orbitope . 76 4.2 Extension property . 80 4.3 Spectrahedral and semidefinite representability of orbitopes un- der the bitorus . 85 4.4 Generalized extension property . 86 5 Orbitopes which are not spectrahedral shadows 92 5.1 How to find orbitopes, which are not spectrahedral shadows . 92 5.2 Orbitopes under the bitorus . 99 5.3 Toric orbitopes . 106 5 5.4 Tensor powers of the standard representation . 108 6 Conclusions and future work 112 7 Zusammenfassung 116 Appendix A Proof of inequality 120 Appendix B List of simple Lie algebras 122 Appendix C List of symbols 140 Bibliography 142 6 Chapter 1 Introduction 1.1 Motivation Orbitopes are highly symmetric objects. Many questions regarding convex sets can be simplified by this symmetry when regarding orbitopes. A simple example of an orbitope is the unit ball in the three-dimensional space. A typical problem is optimizing a linear functional over a given set. Lie groups frequently appear in mathematical problems and physics. Whenever we want to optimize a linear functional over a Lie group, this leads to an optimiza- tion problem over an orbitope. One way to solve such a problem lies in semidefinite programming. The cen- tral objects of semidefinite programming are spectrahedra and their images un- der projections, so called spectrahedral shadows. A semidefinitely representable set is such a spectrahedral shadow. There exist efficient interior point methods, which solve these linear optimization problems over spectrahedra and spectrahe- dral shadows in polynomial time. This motivates us to study the spectrahedral and semidefinite representability of orbitopes. The aim of this thesis is to get a better understanding of which orbitopes are spectrahedra or spectrahedral shadows. 1.2 Acknowledgments I want to thank Prof. Dr. Claus Scheiderer for his support, many illuminating conversations, critical comments and useful remarks. He has been extremely helpful for the research and the creation of this thesis. Professor Scheiderer was always available for questions and it was inspiring working with him. I also thank the DFG, which supported the research with generous fundings. 1.3 Related work The notion of orbitopes was introduced by Sanyal, Sottile and Sturmfels in [38]. An orbitope is the convex hull of an orbit, given by a compact linear algebraic group, acting linearly on a real vector space. Orbitopes can be studied from many different points of view. 7 On the one hand orbitopes are convex and we can ask the typical questions arising from convex geometry. On the other hand the underlying orbit of an orbitope is semialgebraic and being the convex hull of a semialgebraic set, the orbitope is semialgebraic as well (see Proposition 2.6.3). So we can also study orbitopes using techniques from (real) algebraic geometry. Every compact Lie group is isomorphic to a closed linear group ([26] Corol- lary 4.22). This means, whenever we have a linear optimization problem over a compact Lie group, an optimization problem over an orbitope occurs. A spectrahedron is the intersection of the set of positive semidefinite real matrices with an affine linear subspace of the symmetric matrices. The images of spectrahedra under linear projections are semidefinitely representable sets, in short spectrahedral shadows. Every spectrahedral shadow is convex and semi- algebraic. Helton and Nie conjectured that every convex semialgebraic set is a spectrahedral shadow ([22] Section 6). The conjecture is based on the results in [21] and [22], where Helton and Nie use the Lasserre relaxation (see [28]), to essentially show that every convex, semialgebraic set with non-singular bound- ary and strictly positive curvature is a spectrahedral shadow. Scheiderer proved the Helton-Nie conjecture to be true on the plane (see [41]). But only recently Scheiderer disproved the conjecture in [44]. Spectrahedral shadows are at the heart of semidefinite programming (SDP). SDP optimizes a linear functional over a spectrahedron. Since every polyhedron is a spectrahedron, SDP is a generalization of linear programming. It has a wide range of applications such as in control theory, graph theory and many others. There exist efficient interior point methods, which solve SDP problems in polynomial time. In [38] Sanyal et al. pose ten questions regarding orbitopes. Since we offer at least partial results for all those questions, we list them here, to give the reader an overview of problems regarding orbitopes. The questions are essentially (with slight adjustments) the following: We take O to be an orbitope. Q1: Are all faces of O exposed? Faces play a central role in convex geometry. Proper exposed faces are given as intersections of so called supporting hyperplanes with the given convex set. Inclusion gives a partial ordering on the faces and the face lattice is the description of that partial ordering. Q2: What is the face lattice of O? Carath´eodorys theorem (see Theorem 2.2.2) essentially implies that every element v of a compact convex set C is a convex combination of finitely many extreme points. Take nv to be the minimal number to describe v as the convex combination of nv many extreme points. Then the Carath´eodory number of an orbitope O is given as maxfnv : v 2 Og. Q3: What is the Carath´eodory number of O? Whenever the underlying representation of an orbitope is continuous (which we always assume), the orbitope is a semialgebraic set (see Proposition 2.6.3). So from a real algebraic geometry point of view we can ask: Q4: Is O basic semialgebraic? 8 The Zariski closure of the euclidean boundary of an orbitope O is its algebraic boundary and denoted as @a(O). The algebraic boundary of a full-dimensional orbitope is always a hypersurface (see Proposition 2.3.16), so it is given as the vanishing set of a single polynomial. Q5: What is the algebraic boundary @a(O) of O? The polar of a convex set C in V is the set of linear functionals l in the dualspace V ∗, such that l(C) ≤ 1. The polar set of an orbitope is called the coorbitope and denoted as O◦. Q6: What is the algebraic boundary of the coorbitope O◦? The k-th secant variety of an orbitope conv(G · v) is the Zariski closure of the union of all (k +1)-flats spanned by points of the orbit G·v. Secant varieties and the Carath´eodory numbers are connected through Proposition 2.6.7. Q7: Is @aO a secant variety of the orbit? Spectrahedra are basic closed and all their faces are exposed. This means whenever an orbitope is a spectrahedron, Q1 and Q4 are immediately answered positively. Q8: Is O a spectrahedron? One could ask whether every orbitope is a spectrahedron, but Sinn found an example of orbitopes, which have non-exposed faces and as such are not spectrahedra ([47] Corollary 3.1.15). By a result in [38] the examples found by Sinn are known to be spectrahedral shadows. Q9: Is O a spectrahedral shadow? For optimization purposes one is interested in finding SDP representations of small size. As we mentioned before one can optimize linear functionals over spectrahedral shadows efficiently. Q10: What methods can we use to optimize a linear functional over O? In [38] Sanyal et al. answer many of their questions for examples of or- bitopes. Most notably, for this thesis, they prove that every symmetric and skew-symmetric Schur-Horn orbitope and every Fan orbitope is a spectrahe- dron. These are special cases of polar orbitopes, which we discuss in Chapter 3.1. Sanyal et al. also prove that every orbitope under the torus T := fz 2 C : zz = 1g (the group action being the usual multiplication) is a spectrahedral shadow. In Chapter 4 we generalize this result to a wider range of (multivariate) toric orbitopes. Since then other authors have answered some of the posed questions for more classes of orbitopes. Two works about orbitopes, we wish to highlight, are [5] by Biliotti, Ghigi and Heinzner about polar orbitopes and [40] by Saunderson, Parrilo and Willsky about the convex hulls of the special orthogonal groups. In [5] Theorem 3.2 Biliotti et al. prove, that every face of a polar orbitope under a connected Lie group is exposed. Their result answers Q1 positively for these polar orbitopes. The main result of [5] is Theorem 1.1, which gives a bijec- tion between orbits of faces of a polar orbitope and of its so called momentum polytope.