3:00pm - 5:00pmSparsity in polynomial systems and applications
Chair(s): Timo de Wolff (Technische Universität Berlin, Germany), Mareike Dressler (University of California, San Diego)
In this session we bring together researchers working in different areas involving sparsity in applications and sparse polynomial systems. The principle of sparsity is to represent a structure by functions, e.g., polynomials, with as few variables or terms as possible. It is ubiquitous in various areas and problems, where algebra and geometry play a key role. Recently, it has been succesfully applied to problems such as sparse interpolation, polynomial optimization, sparse elimination, fewnomial theory, or tensor decomposition.
This minisymposium provides an opportunity to learn about a selection of these recent developments and explore new potential applications of sparsity.
(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)
Optimal Descartes' rule of signs for polynomial systems supported on circuits
Frédéric Bihan1, Alicia Dickenstein2, Jens Forsgaard3
1Université Savoie Mont Blanc, France, 2Universidad de Buenos Aires, Argentina, 3Universiteit Utrecht, The Netherlands
We will describe a refinement of the Descartes'rule of signs for polynomial systems supported on circuits which wasproposed by Bihan and Dickenstein few years ago. The main difference is that new bound is sharp for any given circuit, and is always smaller or equal to the normalized volume of the convex hull of the circuit. This is a joint work with Alicia Dickenstein and Jens Forsgard.
Polyhedral Approximations to the Cone of Nonnegative Polynomials
Alperen Ergür
Technische Universität Berlin, Germany
Can we always approximate a semidefinite program with a linear program? Is it possible to approximately check nonnegativity of a polynomial by just a few pointwise evaluations? These questions fall into the general framework of approximating the cone of nonnegative polynomials with polyhedral cones.
In this talk, we will show inapproximability of the cone of nonnegative polynomials in the dense case, and existence of a polyhedral approximation with polynomial number of facets in the sparse case (i.e. the case of a subspace of polynomials with fixed dimension). Time permits, we will also discuss a randomized construction of the approximation cone based on a different set of tools coming from computational geometry.
Nonegativity and Discriminants
Jens Forsgaard1, Timo de Wolff2
1Universiteit Utrecht, The Netherlands, 2Technische Universität Berlin, Germany
We study the class of nonnegative polynomials obtained from the inequality of arithmetic and geometric means, called emph{agiforms} or emph{nonnegative circuit polynomials}. They generate a full dimensional subcone $S$ of the cone of all nonnegative polynomials, which is distinct from the cone of sums of squares. Let $mathbb{R}^A$ denote the space of all real polynomials with support $A$. We describe the boundary of the cone $S cap mathbb{R}^A$ as a space stratified in real semi-algebraic varieties. In order to describe the strata, we take a journey through discriminants, polytopes and triangulations, oriented matroids, and tropical geometry. This is based on joint work with Timo de Wolff.
Exploiting Sparsity for Semi-Algebraic Set Volume Computation
Jean-Bernard Lasserre1, Matteo Tacchi1, Tillmann Weisser2, Didier Henrion1
1CNRS-LAAS, Toulouse, France, 2Los Alamos National Lab, NM, USA
We provide a systematic deterministic numerical scheme to approximate the volume (i.e. the Lebesgue measure) of a basic semi-algebraic set whose description follows a sparsity pattern. As in previous works (without sparsity), the underlying strategy is to consider an infinite-dimensional linear program on measures whose optimal value is the volume of the set. This is a particular instance of a generalized moment problem which in turn can be approximated as closely as desired by solving a hierarchy of semidefinite relaxations of increasing size. The novelty with respect to previous work is that by exploiting the sparsity pattern we can provide a sparse formulation for which the associated semidefinite relaxations are of much smaller size. In addition, we can decompose the sparse relaxations into completely decoupled subproblems of smaller size, and in some cases computations can be done in parallel. To the best of our knowledge, it is the first contribution that exploits sparsity for volume computation of semi-algebraic sets which are possibly high-dimensional and/or non-convex and/or non-connected.