3:00pm - 5:00pmAlgebraic methods for convex sets
Chair(s): Rainer Sinn (Freie Universität Berlin, Germany), Greg Blekherman (Georgia Institute of Technology), Daniel Plaumann (Technische Universität Dortmund), Yong Sheng Soh (Institute of High Performance Computing, Singapore), Dogyoon Song (Massachusetts Institute of Technology)
Convex relaxations are extensively used to solve intractable optimization instances in a wide range of applications. For example, convex relaxations are prominently utilized to find solutions of combinatorial problems that are computationally hard. In addition, convexity-based regularization functions are employed in (potentially ill-posed) inverse problems, e.g., regression, to impose certain desirable structure on the solution.
In this mini-symposium, we discuss the use of convex relaxations and the study of convex sets from an algebraic perspective. In particular, the goal of this minisymposium is to bring together experts from algebraic geometry (real and classical), commutative algebra, optimization, statistics, functional analysis and control theory, as well as discrete geometry to discuss recent connections and discoveries at the interfaces of these fields.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
Determinantal representations of stable and hyperbolic polynomials
Victor Vinnikov
Ben Gurion University of the Negev
Positive self-adjoint determinantal representations of homogeneous hyperbolic polynomials certify the hyperbolicity and represent the corresponding hyperbolicity cone as a spectrahedron; they play therefore a key role in convex algebraic geometry. I will talk both about them and about their (non homogeneous) complex cousins --- complex polynomials that are stable with respect to the unit polydisc or the product of upper halfplanes, and the determinantal representations thereof that certify the corresponding stability property
Noncommutative polynomials describing convex sets
Jurij Volcic
Texas A&M University
In their 2012 Annals paper, Helton and McCullough proved that every convex semialgebraic matrix set is described by a linear matrix inequality (LMI). In this talk we first prove that every irreducible noncommutative polynomial $f$ with convex semialgebraic set $D_f = {X: f(X)succ0 }$ must be of degree at most 2 and concave. Furthermore, for a matrix of noncommutative polynomials $F$ we present effective algorithms for checking whether $D_F$ is convex and finding an LMI representation for convex $D_F$. The derivation of these algorithms yields additional features of convex matrix sets that have no counterparts in the commutative theory. Techniques employed include realization theory, noncommutative algebra and semidefinite programming.
This is joint work with Bill Helton, Igor Klep and Scott McCullough.
Semidefinite Programming and Nash Equilibria in Bimatrix Games
Jeffrey Zhang
Princeton University
We explore the power of semidefinite programming (SDP) for finding additive e-approximate Nash equilibria in bimatrix games. We introduce an SDP relaxation for a quadratic programming formulation of the Nash equilibrium (NE) problem and provide a number of valid inequalities to improve the quality of the relaxation. If a rank-1 solution to this SDP is found, then an exact NE can be recovered. We show that for a strictly competitive game, our SDP is guaranteed to return a rank-1 solution. We propose two algorithms based on iterative linearization of smooth nonconvex objective functions whose global minima by design coincide with rank-1 solutions. Empirically, we demonstrate that these algorithms often recover solutions of rank at most two and e close to zero. Furthermore, we prove that if a rank-2 solution to our SDP is found, then a 5/11-NE can be recovered for any game, or a 1/3-NE for a symmetric game. We then show how our SDP approach can address two (NP-hard) problems of economic interest: finding the maximum welfare achievable under any NE, and testing whether there exists a NE where a particular set of strategies is not played. Finally, we show the connection between our SDP and the first level of the Lasserre/sum of squares hierarchy.
Low Rank Tensor Methods in High Dimensional Data Analysis
Ming Yuan
Columbia University
Large amount of multidimensional data in the form of multilinear arrays, or tensors, arise routinely in modern applications from such diverse fields as chemometrics, genomics, physics, psychology, and signal processing among many others. At the moment, our ability to generate and acquire them has far outpaced our ability to effectively extract useful information from them. There is a clear demand to develop novel statistical methods, efficient computational algorithms, and fundamental mathematical theory to analyze and exploit information in these types of data. In this talk, I will review some of the recent progresses and discuss some of the present challenges.