Semi-algebraic Sets Research Articles

Optimization of trigonometric polynomials with crystallographic symmetry and spectral bounds for set avoiding graphs

AbstractWe provide a new approach to the optimization of trigonometric polynomials with crystallographic symmetry. This approach widens the bridge between trigonometric and polynomial optimization. The trigonometric polynomials considered are supported on weight lattices associated to crystallographic root systems and are assumed invariant under the associated reflection group. On one hand the invariance allows us to rewrite the objective function in terms of generalized Chebyshev polynomials of the generalized cosines; On the other hand the generalized cosines parameterize a compact basic semi algebraic set, this latter being given by an explicit polynomial matrix inequality. The initial problem thus boils down to a polynomial optimization problem that is straightforwardly written in terms of generalized Chebyshev polynomials. The minimum is to be computed by a converging sequence of lower bounds as given by a hierarchy of relaxations based on the Hol–Scherer Positivstellensatz and indexed by the weighted degree associated to the root system. This new method for trigonometric optimization was motivated by its application to estimate the spectral bound on the chromatic number of set avoiding graphs. We examine cases of the literature where the avoided set affords crystallographic symmetry. In some cases we obtain new analytic proofs for sharp bounds on the chromatic number while in others we compute new lower bounds numerically.

A note on the Archimedean Positivstellensatz in real algebraic geometry

A variant of the Archimedean Positivstellensatz is proved which is based on Archimedean semirings or quadratic modules of generating subalgebras. It allows one to obtain representations of strictly positive polynomials on compact semi-algebraic sets by means of smaller sets of squares or polynomials. A large number of examples is developed in detail.

Monomial asymptotic polynomials and applications to polynomial optimization problems

The problem of polynomial optimization plays an important role in many fields such as physics, chemistry, and economics. This problem has received research attention from many mathematicians recently. In this paper, we study the polynomial optimization problem over a non-compact semi-algebraic set, for which its constraint set of polynomials G is asymptotic with a finite family of monomials. By changing variables via a suitable monomial mapping, we transform the problem under consideration into the polynomial optimization problem over a compact semi-algebraic feasible set. We then apply the well-known result that the optimal value of a polynomial over a compact semi-algebraic set can be approximated as closely as desired by solving a hierarchy of semi-definite programs and the convergence is finite generically, to obtain results in the general case when the cone C(G) is unimodular. In particular, in the case of polynomials in two variables, we solve the problem quite completely without requiring C(G) to be unimodular

On Łojasiewicz inequalities and the effective Putinar's Positivstellensatz

The representation of positive polynomials on a semi-algebraic set in terms of sums of squares is a central question in real algebraic geometry, which the Positivstellensatz answers. In this paper, we study the effective Putinar's Positivestellensatz on a compact basic semi-algebraic set S and provide a new proof and new improved bounds on the degree of the representation of positive polynomials. These new bounds involve a parameter ε measuring the non-vanishing of the positive function, the constant c and exponent L of a Łojasiewicz inequality for the semi-algebraic distance function associated to the inequalities g=(g1,…,gr) defining S. They are polynomial in c and ε−1 with an exponent depending only on L. We analyse in details the Łojasiewicz inequality when the defining inequalities g satisfy the Constraint Qualification Condition. We show that, in this case, the Łojasiewicz exponent L is 1 and we relate the Łojasiewicz constant c with the distance of g to the set of singular systems.

Polynomial Optimization Over Unions of Sets

AbstractThis paper studies the polynomial optimization problem whose feasible set is a union of several basic closed semialgebraic sets. We propose a unified hierarchy of Moment-SOS relaxations to solve it globally. Under some assumptions, we prove the asymptotic or finite convergence of the unified hierarchy. Special properties for the univariate case are discussed. The application for computing (p, q)-norms of matrices is also presented.

Rings of differentiable semialgebraic functions

Rings of differentiable semialgebraic functions

Spectrahedral shadows and completely positive maps on real closed fields

In this article we develop new methods for exhibiting convex semialgebraic sets that are not spectrahedral shadows. We characterize when the set of nonnegative polynomials with a given support is a spectrahedral shadow in terms of sums of squares. As an application we prove that the cone of copositive matrices of size n\geq5 is not a spectrahedral shadow, answering a question of Scheiderer. Our arguments are based on the model-theoretic observation that any formula defining a spectrahedral shadow must be preserved by every unital \mathbb{R} -linear completely positive map R\to R on a real closed field extension R of \mathbb{R} .

Moment-SoS methods for optimal transport problems

Most common optimal transport (OT) solvers are currently based on an approximation of underlying measures by discrete measures. However, it is sometimes relevant to work only with moments of measures instead of the measure itself, and many common OT problems can be formulated as moment problems (the most relevant examples being Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-Wasserstein distances, barycenters, and Gromov–Wasserstein discrepancies on Euclidean spaces). We leverage this fact to develop a generalized moment formulation that covers these classes of OT problems. The transport plan is represented through its moments on a given basis, and the marginal constraints are expressed in terms of moment constraints. A practical computation then consists in considering a truncation of the involved moment sequences up to a certain order, and using the polynomial sums-of-squares hierarchy for measures supported on semi-algebraic sets. We prove that the strategy converges to the solution of the OT problem as the order increases. We also show how to approximate linear quantities of interest, and how to estimate the support of the optimal transport map from the computed moments using Christoffel–Darboux kernels. Numerical experiments illustrate the good behavior of the approach.

Reducing Nonnegativity over General Semialgebraic Sets to Nonnegativity over Simple Sets

Reducing Nonnegativity over General Semialgebraic Sets to Nonnegativity over Simple Sets

Oddness of the number of Nash equilibria: The case of polynomial payoff functions

In 1971, Wilson (1971) proved that “almost all” finite games have an odd number of mixed Nash equilibria. Since then, several other proofs have been given, but always for mixed extensions of finite games. In this paper, we present a new oddness theorem for large classes of polynomial payoff functions and semi-algebraic sets of strategies. Additionally, we provide some applications to recent models of games on networks such that Patacchini-Zenou's model about juvenile delinquency and conformism (Patacchini and Zenou, 2012), Calvó-Armengol-Patacchini-Zenou's model about social networks in education (Calvó-Armengol et al., 2009), Konig-Liu-Zenou's model about R&D networks (König et al., 2019), Helsley-Zenou's model about social networks and interactions in cities (Helsley and Zenou, 2014).

Combinatorics of Correlated Equilibria

We study the correlated equilibrium polytope PG of a game G from a combinatorial point of view. We introduce the region of full-dimensionality for this class of polytopes and prove that it is a semialgebraic set for any game. Using a stratification via oriented matroids, we propose a structured method for describing the possible combinatorial types of PG , and show that for ( 2 × n ) -games, the algebraic boundary of the stratification is a union of coordinate hyperplanes and binomial hypersurfaces. Finally, we provide a computational proof that there exists a unique combinatorial type of maximal dimension for generic ( 2 × 3 ) -games.

Convergence rates for sums-of-squares hierarchies with correlative sparsity

AbstractThis work derives upper bounds on the convergence rate of the moment-sum-of-squares hierarchy with correlative sparsity for global minimization of polynomials on compact basic semialgebraic sets. The main conclusion is that both sparse hierarchies based on the Schmüdgen and Putinar Positivstellensätze enjoy a polynomial rate of convergence that depends on the size of the largest clique in the sparsity graph but not on the ambient dimension. Interestingly, the sparse bounds outperform the best currently available bounds for the dense hierarchy when the maximum clique size is sufficiently small compared to the ambient dimension and the performance is measured by the running time of an interior point method required to obtain a bound on the global minimum of a given accuracy.

Ellis enveloping semigroups in real closed fields

We introduce the Boolean algebra of d-semialgebraic (more generally, d-definable) sets and prove that its Stone space is naturally isomorphic to the Ellis enveloping semigroup of the Stone space of the Boolean algebra of semialgebraic (definable) sets. For definably connected o-minimal groups, we prove that this family agrees with the one of externally definable sets in the one-dimensional case. Nonetheless, we prove that in general these two families differ, even in the semialgebraic case over the real algebraic numbers. On the other hand, in the semialgebraic case we characterise real semialgebraic functions representing Boolean combinations of d-semialgebraic sets.

Computing the Homology Functor on Semi-algebraic Maps and Diagrams

Computing the Homology Functor on Semi-algebraic Maps and Diagrams

Periods, power series, and integrated algebraic numbers

Periods are defined as integrals of semialgebraic functions defined over the rationals. Periods form a countable ring not much is known about. Examples are given by taking the antiderivative of a power series which is algebraic over the polynomial ring over the rationals and evaluate it at a rational number. We follow this path and close these algebraic power series under taking iterated antiderivatives and nearby algebraic and geometric operations. We obtain a system of rings of power series whose coefficients form a countable real closed field. Using techniques from o-minimality we are able to show that every period belongs to this field. In the setting of o-minimality we define exponential integrated algebraic numbers and show that exponential periods and the Euler constant is an exponential integrated algebraic number. Hence they are a good candiate for a natural number system extending the period ring and containing important mathematical constants.

The Cone of 5×5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$5\ imes 5$$\\end{document} Completely Positive Matrices

We study the cone of completely positive (cp) matrices for the first interesting case n=5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = 5$$\\end{document}. This is a semialgebraic set for which the polynomial equalities and inequlities that define its boundary can be derived. We characterize the different loci of this boundary and we examine the two open sets with cp-rank 5 or 6. A numerical algorithm is presented that is fast and able to compute the cp-factorization even for matrices in the boundary. With our results, many new example cases can be produced and several insightful numerical experiments are performed that illustrate the difficulty of the cp-factorization problem.

Surjective Nash maps between semialgebraic sets

In this work we study the existence of surjective Nash maps between two given semialgebraic sets S and T. Some key ingredients are: the irreducible components Si⁎ of S (and their intersections), the analytic path-connected components Tj of T (and their intersections) and the relations between dimensions of the semialgebraic sets Si⁎ and Tj. A first step to approach the previous problem is the former characterization done by the second author of the images of affine spaces under Nash maps.The core result of this article to obtain a criterion to decide about the existence of surjective Nash maps between two semialgebraic sets is: a full characterization of the semialgebraic subsetsS⊂Rnthat are the image of the closed unit ballB‾mofRmcentered at the origin under a Nash mapf:Rm→Rn. The necessary and sufficient conditions that must satisfy such a semialgebraic set S are: it is compact, connected by analytic paths and has dimensiond≤m.Two remarkable consequences of the latter result are the following: (1) pure dimensional compact irreducible arc-symmetric semialgebraic sets of dimension d are Nash images ofB‾d, and (2) compact semialgebraic sets of dimension d are projections of non-singular algebraic sets of dimension d whose connected components are Nash diffeomorphic to spheres (maybe of different dimensions).

Stability Analysis of Feedback Systems with ReLU Nonlinearities via Semialgebraic Set Representation

This paper is concerned with the stability/instability analysis problem of feedback systems with rectified linear unit (ReLU) nonlinearities. Such feedback systems arise when we model dynamical (recurrent) neural networks (NNs) and NN-driven control systems where all the activation functions of NNs are ReLUs. In this study, we focus on the semialgebraic set representation characterizing the input-output properties of ReLUs. This allows us to employ a novel copositive multiplier in the framework of the integral quadratic constraint and, thus, to derive a new linear matrix inequality (LMI) condition for the stability analysis of the feedback systems. However, the infeasibility of this LMI does not allow us to obtain any conclusion on the system's stability due to its conservativeness. This motivates us to consider its dual LMI. By investigating the structure of the dual solution, we derive a rank condition on the dual variable certificating that the system at hand is unstable. In addition, we construct a hierarchy of dual LMIs allowing for improved instability detection. We illustrate the effectiveness of the proposed approach by several numerical examples.

Real tropicalization and negative faces of the Newton polytope

In this work, we explore the relation between the tropicalization of a real semi-algebraic set S={f1<0,…,fk<0} defined in the positive orthant and the combinatorial properties of the defining polynomials f1,…,fk. We describe a cone that depends only on the face structure of the Newton polytopes of f1,…,fk and the signs attained by these polynomials. This cone provides an inner approximation of the real tropicalization, and it coincides with the real tropicalization if S={f<0} and the polynomial f has generic coefficients. Furthermore, we show that for a maximally sparse polynomial f the real tropicalization of S={f<0} is determined by the outer normal cones of the Newton polytope of f and the signs of its coefficients. Our arguments are valid also for signomials, that is, polynomials with real exponents defined in the positive orthant.

Slicing the Nash equilibrium manifold

This paper uses tools on the structure of the Nash equilibrium correspondence of strategic-form games to characterize a class of fixed-point correspondences, that is, correspondences assigning, for a given parametrized function, the fixed-points associated with each value of the parameter. After generalizing recent results from the game-theoretic literature, we deduce that every fixed-point correspondence associated with a semi-algebraic function is the projection of a Nash equilibrium correspondence, and hence its graph is a slice of a projection, as well as a projection of a slice, of a manifold that is homeomorphic, even isotopic, to a Euclidean space. As a result, we derive an illustrative proof of Browder’s theorem for fixed-point correspondences.