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).
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Session Overview | |
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Location: Unitobler, F022 104 seats, 126m^2 |
| Date: Tuesday, 09/Jul/2019 | |
| 10:00am - 12:00pm | Room free |
| Unitobler, F022 | |
| 3:00pm - 5:00pm | MS195, part 1: Algebraic methods for convex sets |
| Unitobler, F022 | |
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3:00pm - 5:00pm
Algebraic methods for convex sets 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) The slack variety of a polytope The slack variety of a polytope is an algebraic model for the realization space of a combinatorial class of a polytope. We establish a correspondence between realizations of a given polytope and points in the positive part of a variety of matrices of constrained rank. This allows us to apply the tools of computational algebra to a number of problems in polytope theory, such as rational realizability, projectively uniqueness, non-prescribability of faces and realizability of combinatorial polytopes. We then discuss the relationship between slack varieties, Grassmannians and Gale transforms. Spectrahedral representations of polar orbitopes Let $G$ be a connected compact Lie group. A linear representation $V$ of $G$ with $G$-invariant inner product is called polar if there is a linear subspace $Ssubset V$ that intersects every $G$-orbit orthogonally. A $G$-orbitope in $V$ is the convex hull of a $G$-orbit in $V$. We show that every orbitope in a polar representation of $G$ is a spectrahedron, and we construct an explicit spectrahedral representation. By analyzing the moment polytope we can often reduce the size of this representation. In particular, we arrive at new examples where the representation constructed has minimal size. (Joint work with Tim Kobert.) Sums of squares and quadratic persistence How does one effectively recognize sums of squares? We will focus on new bounds on the number of terms in a sum-of-squares representation for a quadratic form on a real projective subvariety. This talk is based on joint work with G. Blekherman, R. Sinn, and M. Velasco. Semialgebraic Vision In this talk I will discuss examples of problems in computer vision that are inherently semialgebraic but have not been studied from that angle thus far. This leads to interesting gaps between algebraic results and their semialgebraic versions. |
| Date: Wednesday, 10/Jul/2019 | |
| 10:00am - 12:00pm | MS130, part 1: Polynomial optimization and its applications |
| Unitobler, F022 | |
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10:00am - 12:00pm
Polynomial optimization and its applications The importance of polynomial (aka semi-algebraic) optimization is highlighted by the large number of its interactions with different research domains of mathematical sciences. These include, but are not limited to, automatic control, combinatorics, and quantum information. The mini-symposium will focus on the development of methods and algorithms dedicated to the general polynomial optimization problem. Both the theoretical and more applicative viewpoints will be covered. (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) The Geometry of SDP-Exactness in Quadratic Optimization Consider the problem of minimizing a quadratic objective subject to quadratic equations. We study the semialgebraic region of objective functions for which this problem is solved by its semidefinite relaxation. For the Euclidean distance problem, this is a bundle of spectrahedral shadows surrounding the given variety. We characterize the algebraic boundary of this region and we derive a formula for its degree. This is joint work with Corey Harris and Bernd Sturmfels. Semidefinite representations of the set of separable states The convex set of separable states plays a fundamental role in quantum information theory and corresponds to the set of non-entangled states. In this talk I will discuss the question of (exact) semidefinite representations for this convex set. Using connections with nonnegative polynomials and sums of squares I will characterize the cases when this set has, or not, an SDP representation. Noncommutative polynomial optimization and quantum graph parameters Graph parameters such as the stability number and chromatic number can be formulated in several ways. For example as polynomial optimization problems or using nonlocal games (in which two separated parties must convince a referee that they have a valid stable set/coloring of the graph of a certain size). After recalling these formulations, we show how they can be used in quantum information theory to study the power of entanglement. The formulation in terms of nonlocal games gives rise to quantum versions of these graph parameters. The polynomial optimization perspective provides hierarchies of semidefinite programming bounds on the classical parameters and we show how the framework of noncommutative polynomial optimization can be used to obtain analogous hierarchies on the quantum graph parameters. This approach unifies several existing bounds on the quantum graph parameters. On Convexity of Polynomials over a Box In the first and main part of this talk, I show that unless P=NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can test whether a cubic polynomial is convex over a box. This result is minimal in the degree of the polynomial and in some sense justifies why convexity detection in nonlinear optimization solvers is limited to quadratic functions or functions with special structure. As a byproduct, the proof shows that the problem of testing whether all matrices in an interval family are positive semidefinite is strongly NP-hard. This problem, which was previously shown to be (weakly) NP-hard by Nemirovski, is of independent interest in the theory of robust control. I will explain the differences between weak and strong NP-hardness clearly and show how our proof bypasses a step in Nemirovski's reduction that involves "matrix inversion". Indeed, while this operation takes polynomial time, it can result in an exponential increase in the numerical value of the rational numbers involved. In the second and shorter part of the talk, I present sum-of-squares-based semidefinite relaxations for detecting or imposing convexity of polynomials over a box. I do this in the context of the convex regression problem in statistics. I also show the power of this semidefinite relaxation in approximating any twice continuously differentiable function that is convex over a box. |
| 3:00pm - 5:00pm | MS195, part 2: Algebraic methods for convex sets |
| Unitobler, F022 | |
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3:00pm - 5:00pm
Algebraic methods for convex sets 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 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 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. Semidefinite Programming and Nash Equilibria in Bimatrix Games 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 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. |
| Date: Thursday, 11/Jul/2019 | |
| 10:00am - 12:00pm | MS130, part 2: Polynomial optimization and its applications |
| Unitobler, F022 | |
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10:00am - 12:00pm
Polynomial optimization and its applications The importance of polynomial (aka semi-algebraic) optimization is highlighted by the large number of its interactions with different research domains of mathematical sciences. These include, but are not limited to, automatic control, combinatorics, and quantum information. The mini-symposium will focus on the development of methods and algorithms dedicated to the general polynomial optimization problem. Both the theoretical and more applicative viewpoints will be covered. (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) Moments and convex optimization for analysis and control of nonlinear partial differential equations This talk will present a convex-optimization-based framework for analysis and control of nonlinear partial differential equations. The approach uses a particular weak embedding of the nonlinear PDE, resulting in a linear equation in the space of Borel measures. This equation is then used as a constraint of an infinite-dimensional linear programming problem (LP). This LP is then approximated by the classical Lasserre hierarchy of convex, finite-dimensional, semidefinite programming problems (SDPs). In the case of analysis of uncontrolled PDEs, the solutions to these SDPs provide bounds on a specified, possibly nonlinear, functional of the solutions to the PDE; in the case of PDE control, the solutions to these SDPs provide bounds on the optimal value of a given optimal control problem as well as suboptimal feedback controllers. The entire approach is based solely on convex optimization with no reliance on spatio-temporal gridding. The approach is applicable to a very broad class of fully nonlinear PDEs with polynomial data. Two-player games between polynomial optimizers and semidefinite solvers. We interpret some wrong results, due to numerical inaccuracies, already observed when solving semidefinite programming (SDP) relaxations for polynomial optimization, on a double precision floating point solver. It turns out that this behavior can be explained and justified satisfactorily by a relatively simple paradigm. In such a situation, the SDP solver - and not the user - performs some "robust optimization" without being told to do so. In other words the resulting procedure can be viewed as a "max-min" robust optimization problem with two players: the solver which maximizes on a ball of arbitrary small radius, centered at the input polynomial, and the user who minimizes over the original decision variables. Next, we consider the problem of finding exact sums of squares (SOS) decompositions for certain classes of polynomials, while relying on arbitrary-precision SDP solvers. We provide a perturbation-compensation algorithm computing exact decompositions for polynomials lying in the interior of the SOS cone. First, the user perturbates the input polynomial to obtain an approximate SOS decomposition. Then, one obtains an exact SOS decomposition after compensating the numerical error with the perturbation terms. We prove that this algorithm runs in boolean time, which is polynomial in the degree of the input and simply exponential in the number of variables. We apply this algorithm to compute exact Polya and Putinar's representations, respectively for positive definite forms and positive polynomials over basic compact semi-algebraic sets. A Generalization of SAGE Certificates for Constrained Optimization We describe a generalization of the SAGE relaxation methodology for obtaining bounds on constrained signomial and polynomial optimization problems. Our approach leverages the fact that SAGE conveniently and transparently blends with convex duality, in a manner that SOS does not. Key properties of traditional SAGE relaxations (e.g. sparsity preservation) are retained by this more general approach. We illustrate the utility of this methodology with a range of examples from the global optimization literature, along with a publicly available software package. On positive duality gaps in semidefinite programming We present a novel analysis of semidefinite programs (SDPs) with positive duality gaps, i.e., different optimal values in the primal and dual problems. These SDPs are considered extremely pathological, they are often unsolvable, and they also serve as models of more general pathological convex programs. We first characterize two variable SDPs with positive gaps: we transform them into a standard form which makes the positive gap easy to recognize. The transformation is very simple, as it mostly uses elementary row operations coming from Gaussian elimination. We next show that the two variable case sheds light on larger SDPs with positive gaps: we present SDPs in any dimension in which the positive gap is certified by the same structure as in the two variable case. We analyze an important parameter, the singularity degree of the duals of our SDPs and show that it is the largest that can result in a positive gap. We complete the paper by generating a library of difficult SDPs with positive gaps (some of these SDPs have only two variables), and a computational study. |
| 3:00pm - 5:00pm | MS195, part 3: Algebraic methods for convex sets |
| Unitobler, F022 | |
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3:00pm - 5:00pm
Algebraic methods for convex sets 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) Average-Case Algorithm Design Using Sum-of-Squares Finding planted "signals" in random "noise" is a theme that captures problems arising in several different areas such as machine learning (compressed sensing, matrix completion, sparse principal component analysis, regression, recovering planted clusters), average-case complexity (stochastic block model, planted clique, random constraint satisfaction), and cryptography (attacking security of pseudorandom generators). For some of these problems (e.g. variants of compressed sensing and matrix completion), influential works in the past two decades identified the right convex relaxation and techniques for analyzing it that guarantee nearly optimal (w.r.t. to the information theoretic threshold) recovery guarantees. However these methods are problem specific and do not immediately generalize to other problems/variants. Fitting Semidefinite-Representable Sets to Support Function Evaluations The geometric problem of estimating an unknown convex set from its support function evaluations arises in a range of scientific and engineering applications. Traditional approaches typically rely on estimators that minimize the error over all possible compact convex sets. These methods, however, do not allow for the incorporation of prior structural information about the underlying set and the resulting estimates become increasingly more complicated to describe as the number of measurements available grows. We address these shortcomings by describing and analyzing a framework based on searching over structured families of convex sets that are specified as linear images of the free spectrahedron. Our results highlight the utility of our framework in settings where the number of measurements available is limited and where the underlying set to be reconstructed is non-polyhedral. A by-product of our framework that arises from taking the appropriate dual perspective is a numerical tool for computing the optimal approximation of a given convex set as a spectrahedra of fixed size. Measuring Optimality Gap in Conic Programming Approximations with Gaussian Width It is a common practice to approximate hard optimization problems with simpler convex programs for the purpose of computational efficiency. However, this often introduces a nontrivial optimality gap between the true optimum and the approximate values. We evaluate the quality of approximations by studying the Gaussian width of the underlying convex cones as a generic measure to evaluate the optimality gap. Specifically, we consider two canonical examples: (a) approximation of the positive semidefinite (PSD) cone by the (scaled) diagonally dominant (DD) cone ($$DD^n, SDD^n$$); and (b) the sequence of hyperbolic cones, $$mathbb{R}^{n,(k)}$$, which are the derivative relaxations of the nonnegative orthant. We show that there is a significant gap between the width of PSD cone and (S)DD cone ($$Theta(n^2)$$ vs $$Theta(n log n)$$). On the other hand, (perhaps, surprisingly) the width of the hyperbolic cones remains almost invariant in the linear regime of relaxation ($$k = alpha n$$for $$alpha < 1$$). False discovery and its control for low rank estimation Cross-Validation (CV) is a commonly employed procedure that selects a model based on predictive evaluations. Despite its widespread use, empirical and theoretical studies have shown that CV produces overly complex models that consist of many false detections. As such, decades of research in statistics has lead to model selection techniques that assess the extent to which the estimated model signifies discoveries about an underlying phenomena. However, existing approaches rely on the discrete structure of the decision space and are not applicable in settings where the underlying model exhibits a more complicated structure such as low-rank estimation problems. We address this challenge via a geometric reformulation of the concept of true/false discovery, which then enables a natural definition in the low-rank case. We describe and analyze a generalization of the Stability Selection method of Meinshausen and Buehlmann to control for false discoveries in low-rank estimation, and we demonstrate its utility via numerical experiments. Concepts from algebraic geometry (e.g. tangent spaces to determinantal varieties) play a central role in the proposed framework. |
| Date: Friday, 12/Jul/2019 | |
| 10:00am - 12:00pm | MS130, part 3: Polynomial optimization and its applications |
| Unitobler, F022 | |
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10:00am - 12:00pm
Polynomial optimization and its applications The importance of polynomial (aka semi-algebraic) optimization is highlighted by the large number of its interactions with different research domains of mathematical sciences. These include, but are not limited to, automatic control, combinatorics, and quantum information. The mini-symposium will focus on the development of methods and algorithms dedicated to the general polynomial optimization problem. Both the theoretical and more applicative viewpoints will be covered. (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) Limitations on the expressive power of convex cones without long chains of faces Recently Averkov showed that various convex cones related to nonnegative polynomials do not have K-lifts (representations as projections of linear sections of another convex cone K) where K is a Cartesian product of positive semidefinite cones of "small" size. In this talk I'll explain how to extend Averkov's approach to show that cones with certain neighborliness properties do not have K-lifts whenever K is a Cartesian product of cones, each of which does not have any long chains of faces (such as smooth cones, low-dimensional cones, and cones defined by hyperbolic polynomials of low degree). On the exactness of Lasserre relaxations and pure states over real closed fields Consider a finite system of non-strict real polynomial inequalities and suppose its solution set $SsubseteqR^n$ is convex, has nonempty interior and is compact. Suppose that the system satisfies the Archimedean condition, which is slightly stronger than the compactness of $S$. Suppose that each defining polynomial satisfies a second order strict quasiconcavity condition where it vanishes on $S$ (which is very natural because of the convexity of $S$) or its Hessian has a certain matrix sums of squares certificate for negative-semidefiniteness on $S$ (fulfilled trivially by linear polynomials). Then we show that the system possesses an exact Lasserre relaxation. High-dimensional estimation via sum-of-squares proofs Estimation is the computational task of approximately recovering a hidden parameter x associated with a distribution D_x given a draw y from the distribution D_x. Numerous interesting questions in statistics, machine learning, and signal processing are captured in this way, for example, sparse linear regression, Gaussian mixture models, topic models, and stochastic block models. In many cases, there is currently a large gap between the statistical guarantees of computationally efficient algorithms and the guarantees of computationally inefficient methods; it is an open question if this gap is inherent in these cases or if better computationally efficient estimation algorithms exist. In this talk, I will present a meta-algorithm for estimation problems based on the sum-of-squares method of Shor, Parrilo, and Lasserre. For some problems, e.g., learning mixtures of spherical Gaussians, this meta-algorithm is able to close previous long-standing gaps and achieve nearly optimal statistical guarantees. Furthermore, it is plausible that, for a wide range of estimation problems, the statistical guarantees that this meta-algorithm achieves are best possible among all efficient algorithms. This talk is based on an ICM proceedings article with Prasad Raghavendra and Tselil Schramm. Exact Optimization via Sums of Nonnegative Circuits and Sums of AM/GM ExponentialsLog-concave polynomials, entropy, and approximate counting We provide two hybrid numeric-symbolic optimization algorithms, computing exact sums of nonnegative circuits (SONC) and sums of arithmetic-geometric-exponentials (SAGE) decompositions. Moreover, we provide a hybrid numeric-symbolic decision algorithm for polynomials lying in the interior of the SAGE cone. Each framework, inspired by previous contributions of Parrilo and Peyrl, is a rounding-projection procedure. For a polynomial lying in the interior of the SAGE cone, we prove that the decision algorithm terminates within a number of arithmetic operations, which is polynomial in the degree and number of terms of the input, and singly exponential in the number of variables. We also provide experimental comparisons regarding the implementation of the two optimization algorithms. |
| 3:00pm - 5:00pm | MS129, part 1: Sparsity in polynomial systems and applications |
| Unitobler, F022 | |
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3:00pm - 5:00pm
Sparsity in polynomial systems and applications 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 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 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 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 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. |
| Date: Saturday, 13/Jul/2019 | |
| 10:00am - 12:00pm | MS130, part 4: Polynomial optimization and its applications |
| Unitobler, F022 | |
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10:00am - 12:00pm
Polynomial optimization and its applications The importance of polynomial (aka semi-algebraic) optimization is highlighted by the large number of its interactions with different research domains of mathematical sciences. These include, but are not limited to, automatic control, combinatorics, and quantum information. The mini-symposium will focus on the development of methods and algorithms dedicated to the general polynomial optimization problem. Both the theoretical and more applicative viewpoints will be covered. (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) Tighter bounds through rank-one convexification To be completed.In classical polynomial optimization, point evaluation is relaxed to integration with respect to a measure. By optimizing over a measure instead of points, the search for the global optimum becomes a linear problem on a measure, and thanks to results from real algebraic geometry, a conic program on moments. The most common approach of this kind is the Lasserre (or SDP) hierarchy where linear constraints on the moments are combined with positive semi-definite constraints on the moment matrix. In case bounds on the optimization variables of the original problem are known, we propose to strengthen the SDP hierarchy by adding non-linear moment constraints that are derived from necessary condition on the moment matrix to be rank-one. Though non-linear, the additional constraints do not change the convex character of the relaxation. Hence, local non-linear solvers are able to solve the tightened relaxation to optimality. Sieve-SDP: a simple facial reduction algorithm to preprocess semidefinite programs We introduce Sieve-SDP, a simple algorithm to preprocess semidefinite programs (SDPs). Sieve-SDP belongs to the class of facial reduction algorithms. It inspects the constraints of the problem, deletes redundant rows and columns, and reduces the size of the variable matrix. It often detects infeasibility. It does not rely on any optimization solver: the only subroutine it needs is Cholesky factorization, hence it can be implemented in a few lines of code in machine precision. We present extensive computational results on more than seven hundred SDPs from the literature. We conclude that Sieve-SDP is very fast: preprocessing by Sieve-SDP takes, on average, less than one percent of the time that it takes for a commercial solver to solve an SDP. Preprocessing by Sieve-SDP significantly improves accuracy and reduces the solution time. Phaseless rank The phaseless rank of a nonnegative matrix M is defined to be the least k for which there exists a complex rank-k matrix N such that |N|=M, entrywise speaking. In optimization terms, it is the solution to the rank minimization of a matrix under complete phase uncertainty on the entries. This concept has a strong connection not only with amoebas, since computing the phaseless rank amounts to solving the amoeba membership problem for the varieties of matrices with restricted rank, but also with positive semidefinite matrix factorizations, as the phaseless rank can be used to derive bounds on the psd rank. In 1966, Camion and Hoffman proved a landmark characterization of what can be seen as the set of singular square matrices with respect to phaseless rank. In this talk we will revisit and extend these results, and show how they can be used to derive some new examples in both amoeba theory and psd factorizations. Log-concave polynomials, entropy, and approximate counting A polynomial is called completely log-concave if it is log-concave as a function on the nonnegative orthant and its directional derivatives have the same property. This class of polynomials includes homogeneous stable polynomials and the basis generating polynomials of matroids. I will introduce some of the basic properties of these polynomials and discuss a concave program that based on entropy maximization that can approximate the size of its support. An important application will be approximating the number of bases of a matroid. This is joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan. |
| 3:00pm - 5:00pm | MS129, part 2: Sparsity in polynomial systems and applications |
| Unitobler, F022 | |
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3:00pm - 5:00pm
Sparsity in polynomial systems and applications 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) Filling a much-needed gap in the literature Many of the early examples in the study of psd forms which are not sos (such as the Motzkin form) arise from monomial substitution into the arithmetic-geometric inequality. Thirty years ago, the speaker gave a necessary and sufficient condition for such a form to be sos or not ({it Math. Ann.,} 283 (1989), 431-464 (MR 90i.11043)). He also announced that certain results would appear as part of the "in preparation" paper "Midpoint polytopes and the map $x_i mapsto x_i^k$". That paper never appeared, and this talk is an attempt to reconstruct the missing material. Computing elimination ideals of likelihood equations We develop a probabilistic algorithm for computing elimination ideals of likelihood equations. We show experimentally that it is far more efficient than directly computing Groebner bases or the known interpolation method for medium to large size models. Furthermore, we deduce discriminants of the elimination ideals, which play a central role in real root classification. In particular, we compute the discriminants of the 3 by 3 matrix model and one Jukes-Cantor model in phylogenetics (with sizes over 30 GB and 8 GB text files, respectively). Nonnegative polynomials and circuit polynomials The concept of sums of nonnegative circuit polynomials (SONC) was introduced as a new certificate of nonnegativity of polynomials, which was proved to be efficient in many cases. It is natural to ask which types of nonnegative polynomials admit SONC decompositions and how big the gap between the PSD cone and the SONC cone is. In this talk, we will consider these questions. Moreover, we clarify an important fact that every SONC polynomial decomposes into a sum of nonnegative circuit polynomials with the same support, which reveals the advantage of SONC decompositions for certifying nonnegativity of sparse polynomials compared with the classical SOS decompositions. An Experimental Classification of Maximal Mediated Sets Maximal mediated sets (MMS), first introduced by Bruce Reznick, arise as a natural structure in the study of nonnegative polynomials supported on circuits. Due to Reznick's, de Wolff's, and Iliman's results, given a nonnegative polynomial $f$ supported on a circuit $C$ with vertex set $Delta$, $f$ is a sum of squares if and only if the non-vertex element of $C$ is in the MMS of $Delta$. In this project, we classify MMS experimentally. As a main theoretical result, we show that an MMS is determined by its underlying lattice. This is joint work with Olivia Röhrig and Timo de Wolff. |
