Real Algebraic Geometry in Convex Optimization by Cynthia Leslie Vinzant A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, BERKELEY Committee in charge: Professor Bernd Sturmfels, Chair Professor David Eisenbud Professor Alper Atamt¨urk Spring 2011 Real Algebraic Geometry in Convex Optimization Copyright 2011 by Cynthia Leslie Vinzant 1 Abstract Real Algebraic Geometry in Convex Optimization by Cynthia Leslie Vinzant Doctor of Philosophy in Mathematics University of California, BERKELEY Professor Bernd Sturmfels, Chair In the past twenty years, a strong interplay has developed between convex optimization and algebraic geometry. Algebraic geometry provides necessary tools to analyze the behavior of solutions, the geometry of feasible sets, and to develop new relaxations for hard non-convex problems. On the other hand, numerical solvers for convex optimization have led to new fast algorithms in real algebraic geometry. In Chapter 1 we introduce some of the necessary background in convex optimization and real algebraic geometry and discuss some of the important results and questions in their intersection. One of the biggest of which is: when can a convex closed semialgebraic set be the feasible set of a semidefinite program and how can one construct such a representation? In Chapter 2, we explore the consequences of an ideal I ⊂ R[x1; : : : ; xn] having a real radical initial ideal, both for the geometry of the real variety of I and as an application to sums of squares representations of polynomials. We show that if inw(I) is real radical for a vector w in the tropical variety, then w is in the logarithmic set of its real variety. We also give algebraic sufficient conditions for w to be in the logarithmic limit set of a more general n semialgebraic set. If, in addition, w 2 (R>0) , then the corresponding quadratic module is stable, which has consequences for problems in polynomial optimization. In particular, if n P 2 inw(I) is real radical for some w 2 (R>0) then R[x1; : : : ; xn] + I is stable. This provides a method for checking the conditions for stability given by Powers and Scheiderer. In Chapter 3, we examine fundamental objects in convex algebraic geometry, such as def- inite determinantal representations and sums of squares, in the special case of plane quartics. A smooth quartic curve in the complex projective plane has 36 inequivalent representations as a symmetric determinant of linear forms and 63 representations as a sum of three squares. These correspond to Cayley octads and Steiner complexes respectively. We present exact algorithms for computing these objects from the 28 bitangents. This expresses Vinnikov quartics as spectrahedra and positive quartics as Gram matrices. We explore the geometry of Gram spectrahedra and discuss methods for computing determinantal representations. Interwoven are many examples and an exposition of much of the 19th century theory of plane quartics. 2 In Chapter 4, we study real algebraic curves that control interior point methods in linear programming. The central curve of a linear program is an algebraic curve specified by linear and quadratic constraints arising from complementary slackness. It is the union of the various central paths for minimizing or maximizing the cost function over any region in the associated hyperplane arrangement. We determine the degree, arithmetic genus and defining prime ideal of the central curve, thereby answering a question of Bayer and Lagarias. These invariants, along with the degree of the Gauss image of the curve, are expressed in terms of the matroid of the input matrix. Extending work of Dedieu, Malajovich and Shub, this yields an instance-specific bound on the total curvature of the central path, a quantity relevant for interior point methods. The global geometry of central curves is studied in detail. Chapter 5 has two parts. In the first, we study the kth symmetric trigonometric mo- ment curve and its convex hull, the Barvinok-Novik orbitope. In 2008, Barvinok and Novik introduce these objects and show that there is some threshold so that for two points on S1 with arclength below this threshold the line segment between their lifts to the curve form an edge on the Barvinok-Novik orbitope and for points with arclength above this threshold, their lifts do not form an edge. They also give a lower bound for this threshold and con- jecture that this bound is tight. Results of Smilansky prove tightness for k = 2. Here we prove this conjecture for all k. In the second part, we discuss the convex hull of a general parametrized curve. These convex hulls can be written as spectrahedral shadows and, as we shall demonstrate, one can compute and effectively describe their faces. i To my family. ii Contents List of Figures iv List of Tables v 1 Introduction 1 1.1 Convexity, Optimization, and Spectrahedra . .1 1.2 Real Algebraic Geometry and Sums of Squares . .4 1.3 Convex Algebraic Geometry . .6 1.3.1 Central Curves . .6 1.3.2 Determinantal Representations and Hyperbolic Polynomials . .7 1.3.3 Convex Hulls of Semialgebraic Sets . .8 1.3.4 Gram Matrices and Sums of Squares . .9 1.3.5 SDPs and Polynomial Optimization . 10 2 Real Radical Initial Ideals 11 2.1 Preliminaries: Initial Ideals and Sums of Squares . 11 2.2 Real Tropical Geometry . 15 2.3 Stability of Sums of Squares modulo an Ideal . 22 2.4 Connections to Compactification . 24 2.5 Connections to Puiseux Series . 26 2.6 Computation . 27 3 Quartic Curves and their Bitangents 31 3.1 History and Motivation . 31 3.2 Computing a Symmetric Determinantal Representation from Contact Curves 35 3.3 Cayley Octads and the Cremona Action . 37 3.4 Sums of Three Squares and Steiner Complexes . 41 3.5 The Gram spectrahedron . 46 3.6 Definite Representations of Vinnikov Quartics . 50 3.7 Real and Definite Determinantal Representations . 54 3.8 A Tropical Example . 59 iii 4 The Central Curve of a Linear Program 62 4.1 Background and Definitions . 62 4.2 Plane Curves . 67 4.3 Concepts from Matroid Theory . 71 4.4 Equations defining the central curve . 74 4.5 The Gauss Curve of the Central Path . 79 4.6 Global Geometry of the Central Curve . 84 4.7 Conclusion . 89 5 Convex Hulls of Curves 93 5.1 The Barvinok-Novik orbitope . 93 5.1.1 Trigonometric Moment Curves . 93 5.1.2 Curves Dipping Behind Facets . 95 5.1.3 Understanding the Facet fxk = 1g .................... 96 5.1.4 Proof of Theorem 5.1.1 . 100 5.1.5 Useful Trigonometric Identities . 101 5.2 The Convex Hull of a Parametrized Curve . 102 5.2.1 Facial structure . 102 5.2.2 The Algebraic Boundary . 104 5.2.3 The Boundary of a Face-Vertex Set . 105 5.2.4 Testing Faces . 106 5.2.5 A Four-Dimensional Example . 107 Bibliography 109 iv List of Figures 1.1 A spectrahedron and a non-exposed vertex of its shadow. .3 1.2 Hyperbolic hypersurfaces bounding spectrahedra. .8 3.1 The Edge quartic and some of its 28 bitangents . 33 3.2 A quartic Vinnikov curve from Example 3.6.1 and four contact cubics . 55 3.3 Degeneration of a Vinnikov quartic into four lines. 60 3.4 A tropical quartic with its seven bitangents. 61 3.5 A tropical quartic with two of its contact cubics . 61 4.1 The central curve of six lines for two choices of the cost function . 63 4.2 The DTZ snake with 6 constraints. On the left, a global view of the polygon and its central curve with the line y2 = 1 appearing as part of the curve. On the right a close-up of the central path and its inflection points. 66 4.3 The degree-6 central path of a planar 7-gon in the affine charts fy0 = 1g and fy2 = 1g. Every line passing through [0 : −b2 : b1] intersects the curve in 6 real points, showing the real curve to be 3 completely-nested ovals. 68 4.4 Correspondence of vertices and analytic centers in the two projections of a primal-dual central curve. Here both curves are plane cubics. 88 4.5 A degree 5 primal central curve in R3 and its degree 4 dual central curve in R2......................................... 91 5.1 Projection of the curve C3 onto the facet fx3 = 1g of its convex hull. The 0 tangent vector C3(t0) + C3(t0) for t0 = 2π=5 is shown in red. 96 5.2 The curve Ck−1 with simplices Pk and Qk for k = 3 (left) and k = 4 (right). 97 π 5.3 Here are two examples of the graphs of fj;k(θ). Note that fj;k( 2k−1 t) has roots f1;:::; 2k − 1gnf2j; 2k − 1 − 2jg, all of multiplicity one. 100 5.4 Edge-vertex set of a curve in R3.......................... 103 5.5 Edge-vertex set of the curve (5.12) in R4..................... 108 v List of Tables 3.1 The six types of smooth quartics in the real projective plane. 33 3.2 Statistics for semidefinite programming over Gram spectrahedra. 50 3.3 The real and definite of LMRs of the six types of smooth quartics. 56 vi Acknowledgments I would first like to thank my advisor, Bernd Sturmfels, for teaching me so much and giving me constant guidance and support. His energy, insights, and enthusiasm are inspiring. It has been a pleasure to work with him and his research group at Berkeley. My thanks go to all of my academic siblings, Dustin Cartwright, Melody Chan, Mar´ıaAng´elicaCueto, Alex Fink, Shaowei Lin, Felipe Rincon, Anne Shiu, and Caroline Uhler, and fellow Berke- ley convex-algebraic-geometers Philipp Rostalski and Raman Sanyal for their knowledge, encouragement, and good company.