Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

 
Session Overview
Location: Unitobler, F021
104 seats, 126m^2
Date: Tuesday, 09/Jul/2019
10:00am - 12:00pmMS182, part 1: Matrix and tensor optimization
Unitobler, F021 
 
10:00am - 12:00pm

Matrix and tensor optimization

Chair(s): Max Pfeffer (Max Planck Institute MiS, Leipzig, Germany), André Uschmajew (Max Planck Institute MiS, Leipzig, Germany)

Matrix and tensor optimization has important applications in the context of modern data analysis and high dimensional problems. Specifically, low rank approximations and spectral properties are of interest. Due to their multilinear parametrization, sets of low rank matrices and tensors form sets with interesting, and sometimes challenging, geometric and algebraic structures. Studying such sets of tensors and matrices in the context of algebraic geometry is therefore not only helpful but also necessary for the development of efficient optimization algorithms and a rigorous analysis thereof. In this respect, the area of matrix and tensor optimization relates to the field applied algebraic geometry by the addressed problems and some of the employed concepts. In this minisymposium, we wish to bring the latest developments in both of these aspects to attention.

 

(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)

 

Tensorized Krylov subspace methods

Daniel Kressner
EPF Lausanne, Switzerland

Tensorized Krylov subspace methods are a versatile tool in numerical linear algebra for addressing large-scale applications that involve tensor product structure. This includes the discretization of high-dimensional PDEs, the solution of linear matrix equations, as well as low-rank updates and Frechet derivatives for matrix functions. This talk gives an overview of such methods, with an emphasis on theoretical properties and their connection to multivariate polynomials.

 

Critical points of quadratic low-rank optimization problems

Bart Vandereycken
University of Geneva, Switzerland

The absence of spurious local minima in certain non-convex minimization problems, e.g. in the context of recovery problems in compressed sensing, has recently triggered much interest due to its important implications on the global convergence of optimization algorithms. One example is low-rank matrix sensing under rank restricted isometry properties. It can be formulated as a minimization problem for a quadratic cost function constrained to a low-rank matrix manifold, with a positive semidefinite Hessian acting like a perturbation of identity on cones of low-rank matrices. We present an approach to show strict saddle point properties and absence of spurious local minima for such problems under improved conditions on the restricted isometry constants. This is joint work with André Uschmajew.

 

Matrix product states from an algebraic geometer’s point of view

Tim Seynnaeve
Max Planck Institute MiS, Leizpig, Germany

Matrix product states and uniform matrix product states play a crucial role in quantum physics and quantum chemistry. They are used, for instance, to compute the eigenstates of the Schrödinger equation. Matrix product states provide a way to represent special tensors in an efficient way and uniform matrix product states are partially symmetric analogs of matrix product states.
We apply methods from algebraic geometry to study uniform matrix product states. Our main results concern the topology of the locus of tensors expressed as uMPS, their defining equations and identifiability. By an interplay of theorems from algebra, geometry and quantum physics we answer several questions and conjectures posed by Critch, Morton and Hackbusch.

 

Computation of the norm of a nonnegative tensor

Antoine Gautier
Saarland University, Saarbruecken, Germany

The norm of a tensor can be computed by finding the maximal eigenvalue of a polynomial mapping. This problem is NP-hard in general. We present a nonlinear generalization of the Perron-Frobenius theorem which guarantees that the norm of tensors with nonnegative entries can be computed with a higher-order variant of the power method. This iterative algorithm has global optimal guarantees and a linear convergence rate. We discuss applications in nonconvex optimization and in the computation of centralities measure of multiplex networks.

 
3:00pm - 5:00pmMS191, part 1: Algebraic and geometric methods in optimization.
Unitobler, F021 
 
3:00pm - 5:00pm

Algebraic and geometric methods in optimization.

Chair(s): Jesus A. De Loera (University of California, Davis, United States of America), Rekha Thomas (University of Washington)

Recently advanced techniques from algebra and geometry have been used to prove remarkable results in Optimization. Some examples of the techniques used are polynomial algebra for non-convex polynomial optimization problems, combinatorial tools like Helly's theorem from combinatorial geometry to analyze and solve stochastic programs through sampling, and using ideal bases to find optimality certificates. Test-set augmentation algorithms for integer programming involving Graver sets for block-structured integer programs, come from concepts in commutative algebra. In this sessions experts will present a wide range of results that illustrate the power of the above mentioned methods and their connections to applied algebra and geometry.

 

(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)

 

Integer optimization from the perspective of subdeterminants

Robert Weismantel
ETH Zurich, Switzerland

For an integer optimization problem (IP), one important data parameter is the maximum absolute value among all square submatrices of the constraint matrix. We present recent developments about this topic.

 

The Minimum Euclidean-Norm Point in a Convex Polytope: Wolfe's Combinatorial Algorithm is Exponential

Jamie Haddock
Dept. Math. UCLA, USA

The complexity of Philip Wolfe's method for the minimum Euclidean-norm point problem over a convex polytope has remained unknown since he proposed the method in 1974. The method is important because it is used as a subroutine for one of the most practical algorithms for submodular function minimization. We present the first example that Wolfe's method takes exponential time. Additionally, we improve previous results to show that linear programming reduces in strongly-polynomial time to the minimum norm point problem over a simplex. This is joint work with J.A. De Loera and L. Rademacher.

 

Matrices of bounded factor width and sums of $k$-nomial squares

Joao Gouveia
University of Coimbra, Portugal

In 2004, Boman et al introduced the concept of factor width of a semidefinite matrix $A$. This is the smallest $k$ for which one can write the matrix as $A=VV^T$ with each column of $V$ containing at most $k$ non-zeros. The cones of matrices of bounded factor width give a hierarchy of inner approximations to the PSD cone. In the polynomial optimization context, a generalized Hankel matrix of a polynomial having factor width k corresponds to the polynomial being a sum of squares where each polynomial being squared has support at most $k$.

This connection has recently been explored by Ahmadi and Majumdar to introduce SDSOS, a sum of squares hierarchy based on sums of binomial squares (sobs), but the study of sobs goes back to Robinson, Choi, Lam and Reznick and ultimately Hurwitz. In this presentation we will prove some results on the geometry of the cones of matrices with bounded factor widths and their duals, and use them to derive new results on the existence of certificates of nonnegativity by sums of k-nomial squares.

Joint work with Mina Saee Bostanabad and Alexander Kovačec

 

A friendly smooth analysis of the Simplex method

Sophie Huiberts
CWI, Amsterdam

The simplex method for linear programming is known for its good performance in practice, although the theoretical worst-case performance is exponential in the input size. The smoothed analysis framework of Spielman and Teng (2001) aims to explain the good practical performance. In our work, we improve on all previous smoothed complexity results for the simplex algorithm on all parameter regimes with a substantially simpler and more general proof. This is joint work with Daniel Dadush. https://arxiv.org/abs/1711.05667
 

Date: Wednesday, 10/Jul/2019
10:00am - 12:00pmMS182, part 2: Matrix and tensor optimization
Unitobler, F021 
 
10:00am - 12:00pm

Matrix and tensor optimization

Chair(s): Max Pfeffer (Max Planck Institute MiS, Leipzig, Germany), André Uschmajew (Max Planck Institute MiS, Leipzig, Germany)

Matrix and tensor optimization has important applications in the context of modern data analysis and high dimensional problems. Specifically, low rank approximations and spectral properties are of interest. Due to their multilinear parametrization, sets of low rank matrices and tensors form sets with interesting, and sometimes challenging, geometric and algebraic structures. Studying such sets of tensors and matrices in the context of algebraic geometry is therefore not only helpful but also necessary for the development of efficient optimization algorithms and a rigorous analysis thereof. In this respect, the area of matrix and tensor optimization relates to the field applied algebraic geometry by the addressed problems and some of the employed concepts. In this minisymposium, we wish to bring the latest developments in both of these aspects to attention.

 

(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)

 

Matrix and Tensor Factorizations with Nonnegativity

Eugene Tyrtyshnikov1, Elena Scherbakova2
1Institute of Numerical Mathematics of Russian Academy of Sciences, Lomonosov Moscow State University, 2Lomonosov Moscow State University

In this talk we survey recent essential developments of the ideas of low-rank matrix approximation and consider their extensions to tensors. The practical importance of the very approach consists in its paradigma of using only small part of matrix entries that allows one to construct a sufficiently accurate appoximation in a fast way for ”big data” matrices that cannot be placed in any available computer memory and are accessed implicitly through calls to a procedure producing any individual entry in demand. We consider how this approach can be used in the cases when we need to maintain nonnegativity of the elements.

 

Decompositions and optimizations of conjugate symmetric complex tensors

Zhening Li
University of Portsmouth, UK

Conjugate partial-symmetric (CPS) tensors are the high-order generalization of Hermitian matrices. As the role played by Hermitian matrices in matrix theory and quadratic optimization, CPS tensors have shown growing interest in tensor theory and optimization, particularly in applications including radar signal processing and quantum entanglement. We study CPS tensors with a focus on ranks, rank-one decompositions and optimizations over the spherical constraint. We prove and propose a constructive algorithm to decompose any CPS tensor into a sum of rank-one CPS tensors. Three types of ranks for CPS tensors are defined and shown to be different in general. This leads to the invalidity of the conjugate version of Comon's conjecture. We then study rank-one approximations and matricizations of CPS tensors. By carefully unfolding CPS tensors to Hermitian matrices, rank-one equivalence can be preserved. This enables us to develop new convex optimization models and algorithms to compute best rank-one approximations of CPS tensors. Numerical experiments from various data are performed to justify the capability of our methods.

 

Chebyshev polynomials and best rank-one approximation ratio

Khazhgali Kozhasov
Max Planck Institute MiS, Leipzig, Germany

We establish a new extremal property of the classical Chebyshev polynomials in the context of the theory of rank-one approximations of tensors. We also give some necessary conditions for a tensor to be a minimizer of the ratio of spectral and Frobenius norms. This is joint work with Andrei Agrachev and André Uschmajew.

 

Optimization methods for computing low rank eigenspaces

André Uschmajew
Max Planck Institute MiS, Leipzig, Germany

We consider the task of approximating the eigenspace belonging to the lowest eigenvalues of a self-adjoint operator on a space of matrices, with the condition that it is spanned by low rank matrices that share a common row space of small dimension. Such a problem arises for example in the DMRG algorithm in quantum chemistry. We propose a Riemannian optimization method based on trace minimization that takes orthogonality and low rank constraints simultaneously into account, and shows better numerical results in certain scenarios compared to other current methods. This is joint work with Christian Krumnow and Max Pfeffer.

 
3:00pm - 5:00pmMS191, part 2: Algebraic and geometric methods in optimization.
Unitobler, F021 
 
3:00pm - 5:00pm

Algebraic and geometric methods in optimization.

Chair(s): Jesus A. De Loera (University of California, Davis, United States of America), Rekha Thomas (University of Washington)

Recently advanced techniques from algebra and geometry have been used to prove remarkable results in Optimization. Some examples of the techniques used are polynomial algebra for non-convex polynomial optimization problems, combinatorial tools like Helly's theorem from combinatorial geometry to analyze and solve stochastic programs through sampling, and using ideal bases to find optimality certificates. Test-set augmentation algorithms for integer programming involving Graver sets for block-structured integer programs, come from concepts in commutative algebra. In this sessions experts will present a wide range of results that illustrate the power of the above mentioned methods and their connections to applied algebra and geometry.

 

(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)

 

Convergence analysis of measure-based bounds for polynomial optimization on the box, ball and sphere

Monique Laurent
CWI, Netherlands

We investigate the convergence rate of a hierarchy of measure-based upper bounds introduced by Lasserre (2011) for the minimization of a polynomial f over a compact set K. These bounds are obtained by searching for a degree 2r sum-of-squares density function h minimizing the expected value of f over K with respect to a given reference measure supported by K.

For simple sets like the box [-1,1]^n, the unit ball and the unit sphere (and natural reference measures including the Lebesgue measure), we show that the convergence rate to the global minimum of f is in O(1/r^2) and that this bound is tight for the minimization of linear polynomials.

Our analysis relies on an eigenvalue reformulation of the bounds and links to extremal roots of orthogonal polynomials, and the tightness result exploits a link to cubature rules.

This is based on joint work with Etienne de Klerk and Lucas Slot.

 

Dynamic programming algorithms for integer programming

Frederich Eisenbrand
EPFL, Switzerland

In this talk, I survey recent progress on the complexity of integer programming in the setting in which it lends itself to dynamic programming approaches. Some of the results are tight under the exponential time hypothesis (ETH). I will also mention open problems. For example, tight results for explicit upper bounds on the variables are however not yet known.

The talk is based on joint work with Robert Weismantel.

 

The support of integer optimal solutions

Timm Oertel
Cardiff University, UK

The support of a vector is the number of nonzero-components. We show that given an integral m x n matrix A, the integer linear optimization problem max{ c^Tx : Ax = b, x>=0, x in Z^n } has an optimal solution whose support is bounded by 2m log(2 sqrt(m) ||A||), where ||A|| is the largest absolute value of an entry of A. Compared to previous bounds, the one presented here is independent on the objective function. We furthermore provide a nearly matching asymptotic lower bound on the support of optimal solutions.

 

New Fourier interpolation formulas and optimization in Euclidean space

Maryna Viazovska
EPFL, Switzerland

Recently we have proven that a radial Schwartz function can be uniquely reconstructed from a certain discrete set of it's values and values of its Fourier transform. Being an interesting phenomenon on its own this interpolation formula allowed us to obtain sharp linear programming bounds in dimensions 8 and 24 and to prove universal optimality of E8 and Leech lattices.

This is joint work with H. Cohn, A. Kumar, Stephen D. Miller, and Danylo Radchenko.

 

Date: Thursday, 11/Jul/2019
10:00am - 12:00pmMS153, part 1: Symmetry in algorithmic questions of real algebraic geometry
Unitobler, F021 
 
10:00am - 12:00pm

Symmetry in algorithmic questions of real algebraic geometry

Chair(s): Cordian Riener (UiT - The Arctic University of Norway, Norway), Philippe Moustrou (UiT - The Arctic University of Norway, Norway)

Symmetry arises quite naturally in many computational problems and from a computational perspective, it allows to reduce the complexity of problems. The mini-symposium aims to presents various instances of computational problems in real algebraic geometry, where symmetry playes an important role.

 

(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)

 

Complete positivity and distance-avoiding sets

Fernando de Oliveira Filho
Technical University of Delft

The completely positive matrices form a subcone of the cone of positive semidefinite matrices over which optimization is NP-hard, but which allows us to provide exact formulations for many combinatorial optimization problems: if, for instance, one replaces positive semidefinite matrices by completely positive matrices in the Lovász theta number, the resulting parameter is equal to the independence number instead of being just an upper bound for it.

The Lovász theta number can be extended to distance graphs on infinite spaces, like the unit distance graph of R^n, whose vertices are all points of R^n and in which two vertices are adjacent if they are at distance 1 from each other. In the case of the unit distance graph, the Lovász theta number provides an upper bound for the maximum density m_1(R^n) that a measurable subset of R^n not containing points at distance 1 can have. I will define the cone of completely positive operators on a Hilbert space and show how it can be used to give an exact formulation for parameters such as m_1(R^n), in analogy with the situation of finite graphs.

This exact formulation can then be used to compute better upper bounds and also to give alternative proofs of results such as the Turing computability of m_1(R^n).

(Joint work with Evan DeCorte and Frank Vallentin.)

 

Kissing number of the hemisphere in dimension 8

Maria Dostert
EPFL Lausanne

The kissing number of spherical caps asks for the maximal number of pairwise non-overlapping unit spheres that can simultaneously touch a central spherical cap in n-dimensional Euclidean space. We consider especially the kissing number of the hemisphere in dimension 8. The kissing number of hemispheres provides less symmetries than the kissing number of unit spheres, which makes the problem more difficult.

The kissing number problem of spheres coincides with the problem of finding a maximal spherical code with minimal angular distance pi/3. The famous configuration of 240 points of unit spheres in dimension 8 given by the root lattice E8, which is an optimal spherical code of minimal angular distance pi/3, is unique up to isometry. From these 240 points we get a configuration on the hemisphere with 183 pairwise non-overlapping unit spheres. Bachoc and Vallentin determined an upper bound of 183.012 using semidefinite optimization, hence the kissing number of the hemisphere in dimension 8 is 183.

Using the semidefinite program of Bachoc and Vallentin we obtained a sharp numerical bound of 183. Based on this floating point solution and the configuration given by the E8 lattice we study uniqueness. This is a joint work with David de Laat and Philippe Moustrou.

 

Pair correlation estimates for the zeros of the zeta function via semidefinite programming

David de Laat
MIT

In this talk I will explain how sum-of-squares characterizations and semidefinite programming can be used to obtain improved bounds for quantities related to zeros of the Riemann zeta function. This is based on Montgomery's pair correlation approach. I will show how this connects to the sphere packing problem, and speculate about future improvements.

Joint work with Andrés Chirre and Felipe Gonçalves.

 

Cut polytopes and minors in graphs

Tim Römer
Universität Osnabrück

The study of cuts in graphs is a very interesting topic in discrete mathematics with relations and applications to many other fields like algebraic geometry, algebraic statistics, commutative algebra and combinatorial optimization. Here we focus on cut polytopes and associated algebraic objects. Special attention is given to the (non-)existence of minors of interest in the underlying graph and algebraic as well as geometric consequences of such a fact.

 
3:00pm - 5:00pmMS158, part 1: Structured sums of squares
Unitobler, F021 
 
3:00pm - 5:00pm

Structured sums of squares

Chair(s): James Saunderson (Monash University, Australia), Mauricio Velasco (Universidad de los Andes)

A description of a nonnegative polynomial as a sum of squares gives a concise proof of its nonnegativity. Computationally, such sum-of-squares decompositions are appealing because we can search for them by solving a semidefinite feasibility problem. This connection means that optimization and decision problems arising in a range of areas, from robotics to extremal combinatorics, can be reformulated as, or approximated with, semidefinite optimization problems.

This minisymposium highlights the roles of various kinds of additional structures, including symmetry and sparsity, in understanding when (structured) sum of squares decompositions do and do not exist. It will also showcase interesting connections between sums of squares and a range of areas, such as extremal combinatorics, dynamical systems and control, and algorithms and complexity theory.

 

(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)

 

Learning dynamical systems with side information

Amir Ali Ahmadi, Bachir El Khadir
Princeton University

In several safety-critical applications, one has to learn the behavior of an unknown dynamical system from noisy observations of a very limited number of trajectories. For example, to autonomously land an airplane that has just gone through engine failure, limited time is available to learn the modified dynamics of the plane before appropriate control action can be taken. Similarly, when a new infectious disease breaks out, few observations are initially available to understand the dynamics of contagion.

In situations of this type where data is limited, it is essential to exploit “side information”---e.g. physical laws or contextual information---to assist the task of learning. We present a mathematical formalism of the problem of learning a dynamical system with side information, where side information can mean a concrete collection of local or global properties of the dynamical system. We show that sum of squares optimization is particularly suited for learning a dynamical system that best agrees with the observations and respects the side information.

Based on joint work with Bachir El Khadir (Princeton).

 

Convergence analysis of measure-based bounds for polynomial optimization on compact sets

Lucas Slot, Monique Laurent
CWI Amsterdam

We investigate the convergence rate of a hierarchy of measure-based upper bounds introduced by Lasserre (2011) for the minimization of a polynomial f over a compact set K. These bounds are obtained by searching for a degree 2r sum-of-squares density function h minimizing the expected value of f over K.

The convergence rate to the global minimum of f over K is known to be in O(1/r^2) for special sets K like the box, ball and sphere, so we consider here general compact sets. We show a convergence rate in O((log r)/r) when K satisfies a minor geometric condition and a rate in O(((log r)/r)^2) when K is a convex body, improving on the current best known bounds for these cases. These results can be refined when making assumptions on the order of f at a global minimizer.

Our analysis relies on combining tools from convex geometry and approximation theory, making use in particular of approximations of the Dirac delta function by fast-decreasing polynomials.

This is based on joint work with Monique Laurent.

 

Sums-of-squares for extremal discrete geometry on the unit sphere

Frank Vallentin
Universität zu Köln

In this talk I will show how one can apply sum-of-squares techniques for various extremal geometric problems on the unit sphere, especially finding thinnest coverings or spherical designs.

 

Computing spectral bounds for geometric graphs via polynomial optimization

Philippe Moustrou
UiT - The Arctic University of Norway

A powerful lower bound on the chromatic number of a finite graph is the spectral bound due to Hoffman, which is related to the eigenvalues of the adjacency matrix of the graph. This bound has been generalized by Bachoc, DeCorte, Oliveira and Vallentin to infinite graphs. In this talk we describe how this bound can be adapted to two particular problems arising from geometry:

- Given a norm what is the largest density of a subset of the n-dimensional real space that does not contain any pair of points such that the norm of their difference is 1? This problem is closely related to the determination of the famous chromatic number of the plane.

- What is the least number of colors needed to color the interiors of the cells of the tessellation associated with the Voronoi cell of a given lattice, in such a way that two cells sharing a facet do not receive the same color?

After introducing these two problems, we show how to compute the spectral bound by solving a polynomial optimization problem using sums of squares.

 

Date: Friday, 12/Jul/2019
10:00am - 12:00pmMS179, part 1: Algebraic methods for polynomial system solving
Unitobler, F021 
 
10:00am - 12:00pm

Algebraic methods for polynomial system solving solving

Chair(s): Mohab Safey El Din (Sorbonne Université, France), Éric Schost (University of Waterloo)

Polynomial system solving is at the heart of computational algebra and computational algebraic geometry. It arises in many applications ranging from computer security and coding theory (where computations must be done over finite fields) and engineering sciences such as chemistry, biology, signal theory or robotics among many others (here computations are done over inifinite domains such as complex or real numbers). The need of reliable algorithms for solving these problems is prominent because of the non-linear nature of the problems we have in hand.

Algebraic methods provide a nice framework for designing efficient and reliable algorithms solving polynoial systems. This mini-symposium will cover many aspects of this topic, including design of symbolic computation algorithms as well as the use of numerical methods in this framework with an emphasis on reliability.

 

(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)

 

Exploiting fast linear algebra in the computation of multivariate relations

Vincent Neiger
Univ. Limoges

We consider the problem of computing multivariate relations in a finite-dimensional setting: for a submodule M of K[x]^n such that Q = K[x]^n/M has finite dimension D as a K-vector space, and given elements f1,..,fm in Q, the problem is to compute relations between the fi's, that is, polynomials (p1,..,pm) in K[x]^m such that p1 f1 + ... + pm fm = 0 in Q.

Assume that the multiplication matrices of the r variables with respect to some basis of Q are known. Then, for any monomial order, we give an algorithm for computing the reduced Gröbner basis of the module of such relations using O(r D^w log(D)) operations in the base field K, where w is the exponent of matrix multiplication. We also show that, under some assumptions, the multiplication matrices can be computed from a Gröbner basis of M within the same complexity bound, leading in particular to a change of monomial order algorithm whose complexity bound is sub-cubic in D.

Contains joint work with Éric Schost and Hamid Rahkooy

 

Certification via squaring-up

Timothy Duff
Georgia Tech

We consider numerical certification of approximate solutions to N polynomial equations in n variables in the case where n < N via passing to a square subsystem. Typically the excess complex solutions of a general squaring-up are all isolated, and their number is given in terms of a birationally-invariant intersection index. This enables certification, via Smale and Shub's alpha-theory, for examples where the intersection index is known a priori, or in cases where it may be calculated algorithmically in terms of an associated Newton-Okounkov body.

joint w/ Frank Sottile (Texas A&M)

 

Efficient and complete certification of roots in solving polynomial systems

Michael Burr
Clemson Univ.

A certified algorithm is an algorithm that provides a proof or certificate of the correctness of its output. A complete algorithm is one that can be correctly implemented on a computer using a bit-based computation mode l. In this talk, I will present recent work on certification methods for solving polynomial systems. I will focus on two paradigms for certifying solutions to polynomial systems: A posteriori approaches take the output of (any) computation and check the correctness of the result. A priori approaches prove that an entire computation is correct, including the output. In particular, I will discuss how interval arithmetic-based methods can be used to make certification algorithms efficient in practice.

 

Reconstruction of an Algebraic Surface from a 2D Projection

Joseph Schicho
Johannes Kepler Univ.

An algebraic surface is given by its equation, the zero set of a polynomial in four homogeneous variables. Its picture under central projection is computed as the zero set of its discriminant: a plane algebraic curve. Can we recover the equation of an algebraic surface by its discriminant? If the surface is nonsingular, then the answer is yes, by a result of d'Almeida. If we allow also "generic" singularities, then the there is sometimes a finite list of possibles. This talk explains the resconstruction method and discusses the ambiguities in the singular case.

 
3:00pm - 5:00pmMS158, part 2: Structured sums of squares
Unitobler, F021 
 
3:00pm - 5:00pm

Structured sums of squares

Chair(s): James Saunderson (Monash University, Australia), Mauricio Velasco (Universidad de los Andes)

A description of a nonnegative polynomial as a sum of squares gives a concise proof of its nonnegativity. Computationally, such sum-of-squares decompositions are appealing because we can search for them by solving a semidefinite feasibility problem. This connection means that optimization and decision problems arising in a range of areas, from robotics to extremal combinatorics, can be reformulated as, or approximated with, semidefinite optimization problems.

This minisymposium highlights the roles of various kinds of additional structures, including symmetry and sparsity, in understanding when (structured) sum of squares decompositions do and do not exist. It will also showcase interesting connections between sums of squares and a range of areas, such as extremal combinatorics, logic, dynamical systems and control, and algorithms and complexity theory.

 

(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)

 

Simple Graph Density Inequalities with no Sum of Squares Proofs

Annie Raymond1, Greg Blekherman2, Mohit Singh2, Rekha Thomas3
1University of Massachusetts Amherst, 2Georgia Institute of Technology, 3University of Washington

Establishing inequalities among graph densities is a central pursuit in extremal combinatorics. One way to certify the nonnegativity of a graph density expression is to write it as a sum of squares. In this talk, we identify a simple condition under which a graph density expression cannot be a sum of squares. Using this result, we prove that the Blakley-Roy inequality does not have a sum of squares certificate when the path length is odd. We also show that the same Blakley-Roy inequalities cannot be certified by sums of squares using a multiplier of the form one plus a sum of squares. These results answer two questions raised by Lovász. Our main tool is used again to show that the smallest open case of Sidorenko's conjectured inequality cannot be certified by a sum of squares. This is joint work with Greg Blekherman, Mohit Singh and Rekha Thomas.

 

Symmetry and Nonnegativity

Greg Blekherman
Georgia Institute of Technology

I will discuss several recent results on symmetric nonnegative polynomials and their approximations by sums of squares. I will consider several types of symmetry, but the situation is especially interesting in the limit as the number of variables tends to infinity. There are diverse applications to quantum entanglement, graph density inequalities and theoretical computer science.

 

Symmetry and the Sum of Squares Hierarchy

Aaron Potechin
University of Chicago

In this talk, I will describe how symmetry can be used both to greatly reduce the size of the semidefinite program needed to implement SOS and to make proving SOS lower bounds considerably easier. In particular, I will describe how to construct semidefinite programs for SOS on symmetric problems whose size is independent of n using Razborov's flag algebras. I will also describe a sufficient condition for proving SOS lower bounds when the problem is symmetric.

As an application, I will describe SOS lower bounds for the following Turan-type problem: What is the minimum possible number of triangles in a graph G with n vertices? More generally, I will describe how we can obtain SOS lower bounds almost automatically whenever our problem is symmetric and the difficulty of our problem comes from integrality arguments (i.e. arguments that a certain expression must be an integer).

Note: This talk will be based on the paper "Symmetric Sums of Squares over k-Subset Hypercubes" by Raymond, Saunderson, Singh, and Thomas and my paper "Sum of Squares Lower Bounds from Symmetry and a Good Story".

 

Date: Saturday, 13/Jul/2019
10:00am - 12:00pmMS179, part 2: Algebraic methods for polynomial system solving
Unitobler, F021 
 
10:00am - 12:00pm

Algebraic methods for polynomial system solving solving

Chair(s): Mohab Safey El Din (Sorbonne Université, France), Éric Schost (University of Waterloo)

Polynomial system solving is at the heart of computational algebra and computational algebraic geometry. It arises in many applications ranging from computer security and coding theory (where computations must be done over finite fields) and engineering sciences such as chemistry, biology, signal theory or robotics among many others (here computations are done over inifinite domains such as complex or real numbers). The need of reliable algorithms for solving these problems is prominent because of the non-linear nature of the problems we have in hand.

Algebraic methods provide a nice framework for designing efficient and reliable algorithms solving polynoial systems. This mini-symposium will cover many aspects of this topic, including design of symbolic computation algorithms as well as the use of numerical methods in this framework with an emphasis on reliability.

 

(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)

 

On polynomial and regular images of Euclidean spaces

José Fernando Galvan
Univ. Madrid

Let $f:=(f_1,ldots,f_m):R^ntoR^m$ be a map. We say that $f$ is em polynomial em if its components $f_k$ are polynomials. The map $f$ is em regular em if its components can be represented as quotients $f_k=frac{g_k}{h_k}$ of two polynomials $g_k,h_k$ such that $h_k$ never vanishes on $R^n$. More generally, the map $f$ is em Nash em if each component $f_k$ is a Nash function, that is, an analytic function whose graph is a {sl semialgebraic set}. Recall that a subset $SssubsetR^n$ is em semialgebraic em if it has a description as a finite boolean combination of polynomial equalities and inequalities. By Tarski-Seidenberg's principle the image of a map whose graph is a semialgebraic set is a semialgebraic set. Consequently, the images of polynomial, regular and Nash maps are semialgebraic sets. In 1990 {em Oberwolfach reelle algebraische Geometrie} week Gamboa proposed a kind of converse problem: em To characterize the semialgebraic sets in $R^m$ that are either polynomial or regular images of some $R^n$em. In the same period Shiota formulated a conjecture that characterizes Nash images of $R^n$, which has been recently proved by the author. In this talk we collect some of our main contributions to these problems and announce some future work concerning polynomial images of the unit closed ball. We have approached our contributions along the last two decades in three directions:

(i) To construct explicitly polynomial and regular maps whose images are the members of large families of semialgebraic sets whose boundaries are piecewise linear.

(ii) To find obstructions to be polynomial/regular images of $R^n$.

(iii) To prove Shiota's conjecture and some relevant consequences.

 

Degree bounds for the sparse Nullstellensatz

Gabriela Jeronimo
Univ. Buenos Aires

We will present new upper bounds both the degrees in Hilbert's Nullstellensatz and for the Noether exponent of a polynomial ideal in terms of mixed volumes of convex sets associated with the supports of a finite family of given generators. Our main results are the first upper bounds valid for arbitrary polynomial systems that distinguish the different supports of the polynomials. In the mixed sparse setting, they can be considerably smaller than previously known bounds. This is joint work with María Isabel Herrero and Juan Sabia (Universidad de Buenos Aires and CONICET, Argentina).

 

Signature-based Möller's algorithm for strong Gröbner bases over PIDs

Thibaut Verron
Johannes Kepler Univ.

Signature-based algorithms have become a standard approach for Gröbner basis computations for polynomial rings over fields, and recent work has focused on extending this technique to coefficients in rings.

Möller introduced in 1988 two algorithms for Gröbner bases over rings: one algorithm computing weak bases over any effective ring, and another computing strong bases if the coefficient ring is a Principal Ideal Domain (PID).

In 2018, we showed that it is possible to augment Möller's weak GB algorithm with signatures, in the case of PIDs. In this work, we show how the same technique can be used for Möller's strong GB algorithm. We prove that the resulting algorithm computes a strong Gröbner basis while ensuring that the signatures do not decrease, and in particular, that no signature drop occurs. Möller's strong GB algorithm requires special care compared to Möller's weak GB algorithm or to the fields case, because of its use of so-called $G$-polynomials whose signatures have to be controlled.

As in the case of fields or Möller's weak GB algorithm, it allows to use additional criteria such as the F5 criterion, which allows the algorithm to compute a Gröbner basis without a reduction to zero in the case of an ideal described by a regular sequence.

Furthermore, we show that Buchberger's coprime and chain criteria can be made compatible with signatures in Möller's strong GB algorithm. This makes use of the syzygy characterization of Gröbner bases, given by Möller's lifting theorem.

These results are supported by a toy implementation of the algorithms in Magma. In particular, Möller's strong GB algorithm does not suffer from the same combinatorial bottleneck as Möller's algorithm, which allowed us to gather experimental data regarding the number of $S$-polynomials computed, reduced and eliminated by each criterion.

(Joint work with Maria Francis)

 

Witness collections and a numerical algebraic geometry toolkit

Jose Rodriguez
Univ. of Wisconsin

A numerical description of an algebraic subvariety of projective space is given by a general linear section, called a witness set. For a subvariety of a product of projective spaces (a multiprojective variety), the corresponding numerical description is given by a witness set collection, whose structure is more involved. In this talk we build on recent work to give a complete treatment of witness set collections for multiprojective varieties, together with an algorithm for their numerical irreducible decomposition that exploits the structure of a witness set collection.

 
3:00pm - 5:00pmMS153, part 2: Symmetry in algorithmic questions of real algebraic geometry
Unitobler, F021 
 
3:00pm - 5:00pm

Symmetry in algorithmic questions of real algebraic geometry

Chair(s): Cordian Riener (UiT - The Arctic University of Norway, Norway), Philippe Moustrou (UiT - The Arctic University of Norway, Norway)

Symmetry arises quite naturally in many computational problems and from a computational perspective, it allows to reduce the complexity of problems. The mini-symposium aims to presents various instances of computational problems in real algebraic geometry, where symmetry playes an important role.

 

(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)

 

Orbit closures in the Zariski spectrum of the infinite polynomial ring

Mario Kummer
TU Berlin

We study orbit closures of zero sets of prime ideals under the action of the infinite symmetric group. This leads, for instance, to a characterization of invariant prime ideals of the polynomial ring with n sets of infinitely many variables in terms of families of subvarieties of affine n-space. We also discuss semi-algebraic aspects over the real numbers. This is joint work with Cordian Riener.

 

Sum-of-squares hierarchy for symmetric formulations.

Adam Kurpisz
ETH Zurich

We preset a method for proving Sum-of-Squares (SoS) lower bounds when the initial problem formulation exhibits a high degree of symmetry. Our main technical theorem allows us to reduce the study of the positive semi-definiteness to the analysis of "well-behaved" univariate polynomial inequalities.

We illustrate the technique with few applications. For binary polynomial optimization problems of degree 2d and an odd number of variables n, we prove that frac{n+2d-1}{2} levels of the SoS hierarchy are necessary to provide the exact optimal value. This matches the recent upper bound result by Sakaue, Takeda, Kim and Ito. As a special case we give a short elementary proof of Grigoriev/Laurent lower bound for finding the integer cut polytope of the complete graph. Additionally, we show that the SoS hierarchy requires a non-constant number of rounds to improve the initial integrality gap of 2 for the Min-Knapsack linear program strengthened with cover inequalities.

Finally, we study a conjecture by Laurent, who considered the linear representation of a set with no integral points. She showed that the Sherali-Adams hierarchy requires n levels to detect the empty integer hull, and conjectured that the SoS rank for the same problem is n-1. We disprove this conjecture and derive lower and upper bounds for the rank.

 

Symmetry Preserving Interpolation

Erick Rodriguez Bazan
INRIA

I will talk about multivariate interpolation in the presence of symmetry. Interpolation is a prime tool in algebraic computation while symmetry is a qualitative feature that can be more relevant to a mathematical model than the numerical accuracy of the parameters. In my presentation, I will show how to exactly preserve symmetry in multivariate interpolation while exploiting it to alleviate the computational cost. We revisit minimal degree and least interpolation with symmetry adapted bases, rather than monomial bases. This allows to construct bases of invariant interpolation spaces in blocks, capturing the inherent redundancy in the computations. I will also show that the so constructed symmetry adapted interpolation bases alleviate the computational cost of any interpolation problem and automatically preserve any equivariance of this interpolation problem might have.

 

Separating invariants of finite groups

Fabian Reimers
TU Munich

Let X be an affine variety with an action of an algebraic group G (over an algebraically closed field K). A subset (e.g. a subalgebra) of the invariant ring K[X]^G is called separating if it has the same capability of separating the orbits as the whole invariant ring. In this talk we focus on finite groups and show how the existence of a separating set of small size, or a separating algebra which is a complete intersection, is related to the property of G being a reflection (or bireflection) group. Theorems of Serre, Dufresne, Kac-Watanabe and Gordeev about linear representations are extended to this setting of G-varieties.