UC Berkeley

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 having a real radical initial ideal, both for the geometry its real variety and as an application to sums of squares representations of polynomials. We show that if the initial ideal of an ideal is real radical for a vector in the tropical variety, then this vectors belongs to logarithmic set of its real variety. We also give algebraic sufficient conditions for a ray to be in the logarithmic limit set of a more general semialgebraic set. If, in addition, the ray has positive coordinates, then the corresponding quadratic module is stable, which has consequences for problems in polynomial optimization. In particular, if an ideal has a is real radical initial ideal for some positive weight vector, then the preorder generated by the ideal 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 definite 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.

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 k^th symmetric trigonometric moment 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 S^1 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 conjecture 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.