Semi-algebraic Sets Research Articles

BullsEye : Scalable and Accurate Approximation Framework for Cache Miss Calculation

For Affine Control Programs or Static Control Programs (SCoP), symbolic counting of reuse distances could induce polynomials for each reuse pair. These polynomials along with cache capacity constraints lead to non-affine (semi-algebraic) sets; and counting these sets is considered to be a hard problem. The state-of-the-art methods use various exact enumeration techniques relying on existing cardinality algorithms that can efficiently count affine sets. We propose BullsEye , a novel, scalable, accurate, and problem-size independent approximation framework. It is an analytical cache model for fully associative caches with LRU replacement policy focusing on sampling and linearization of non-affine stack distance polynomials. First, we propose a simple domain sampling method that can improve the scalability of exact enumeration. Second, we propose linearization techniques relying on Handelman’s theorem and Bernstein’s representation . To improve the scalability of the Handelman’s theorem linearization technique, we propose template (Interval or Octagon) sub-polyhedral approximations. Our methods obtain significant compile-time improvements with high-accuracy when compared to HayStack on important polyhedral compilation kernels such as nussinov , cholesky , and adi from PolyBench , and harris , gaussianblur from LLVM -TestSuite. Overall, on PolyBench kernels, our methods show up to 3.31× (geomean) speedup with errors below ≈ 0.08% (geomean) for the octagon sub-polyhedral approximation.

Sum-of-Squares Hierarchies for Polynomial Optimization and the Christoffel--Darboux Kernel

Consider the problem of minimizing a polynomial $f$ over a compact semialgebraic set $\mathbf{X} \subseteq \mathbb{R}^n$. Lasserre introduces hierarchies of semidefinite programs to approximate this hard optimization problem, based on classical sum-of-squares certificates of positivity of polynomials due to Putinar and Schmüdgen. When $\mathbf{X}$ is the unit ball or the standard simplex, we show that the hierarchies based on the Schmüdgen-type certificates converge to the global minimum of $f$ at a rate in $O(1/r^2)$, matching recently obtained convergence rates for the hypersphere and hypercube $[-1,1]^n$. For our proof, we establish a connection between Lasserre's hierarchies and the Christoffel--Darboux kernel, and make use of closed form expressions for this kernel derived by Xu.

Gram spectrahedra of ternary quartics

The Gram spectrahedron of a real form f∈R[x_]2d parametrizes all sum of squares representations of f. It is a compact, convex, semi-algebraic set, and we study its facial structure in the case of ternary quartics, i.e. f∈R[x,y,z]4. We show that the Gram spectrahedron of every smooth ternary quartic has faces of dimension 2, and generically none of dimension 1, thus answering a question in Plaumann et al. (2011) about the existence of positive dimensional faces on such Gram spectrahedra. We complete the proof in Plaumann et al. (2011) showing that the so called Steiner graph of every smooth quartic is isomorphic to K4∐K4. Moreover, we show that the Gram spectrahedron of a generic positive semidefinite ternary quartic contains extreme points of all expected ranks.

Dual Certificates and Efficient Rational Sum-of-Squares Decompositions for Polynomial Optimization over Compact Sets

We study the problem of computing weighted sum-of-squares (WSOS) certificates for positive polynomials over a compact semialgebraic set. Building on the theory of interior-point methods for convex optimization, we introduce the concept of dual certificates, which allows us to interpret vectors from the dual of the sum-of-squares cone as rigorous nonnegativity certificates of a WSOS polynomial. Whereas conventional WSOS certificates are alternative representations of the polynomials they certify, dual certificates are distinct from the certified polynomials; moreover, each dual certificate certifies a full-dimensional convex cone of WSOS polynomials. For a theoretical application, we give a short new proof of Powers's theorems on the existence of rational WSOS certificates of positive polynomials. For a computational application, we show that exact WSOS certificates can be constructed from numerically computed dual certificates at little additional cost, without any rounding or projection steps applied to the numerical certificates. We also present an algorithm for computing the optimal WSOS lower bound of a given polynomial along with a rational dual certificate, with a polynomial-time computational cost per iteration and linear rate of convergence.

Smooth points on semi-algebraic sets

Many algorithms for determining properties of real algebraic or semi-algebraic sets rely upon the ability to compute smooth points. In this paper, we present a procedure based on computing the critical points of some well-chosen function that guarantees the computation of smooth points in each bounded connected component of a (real) atomic semi-algebraic set. Our technique is intuitive in principal, performs well on previously difficult examples, and is straightforward to implement using existing numerical algebraic geometry software. The practical efficiency of our approach is demonstrated by solving a conjecture on the number of equilibria of the Kuramoto model for the n=4 case. We also apply our method to design an algorithm to compute the real dimension of algebraic sets, the original motivation for this research. We compare the efficiency of our method to existing methods to compute the real dimension on a benchmark family.

A disintegration of the Christoffel function

We show that the Christoffel function (CF) factorizes (or can be disintegrated) as the product of two Christoffel functions, one associated with the marginal and the another related to the conditional distribution, in the spirit of “the CF of the disintegration is the disintegration of the CFs”. In the proof one uses an apparently overlooked property (but interesting in its own) which states that any sum-of-squares polynomial is the Christoffel function of some linear form (with a representing measure in the univariate case). The same is true for the convex cone of polynomials that are positive on a basic semi-algebraic set. This interpretation of the CF establishes another bridge between polynomials optimization and orthogonal polynomials.

Small Amplitude Periodic Orbits in Three-Dimensional Quadratic Vector Fields with a Zero-Hopf Singularity

We consider some families of three-dimensional quadratic vector fields having a fixed zero-Hopf equilibrium. We are interested in the bifurcation of periodic ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document}-orbits from the singularity, that is, those small amplitude orbits that make a fixed arbitrary number ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} of revolutions about a rotation axis and then returns to the initial point closing the orbit. When the parameters of the family are restricted to certain explicitly computable open semi-algebraic sets Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}, we characterize those parameters that give rise to the appearance of local two-dimensional periodic invariant manifolds through the singularity. Also we use a Bautin-type analysis to study the maximum number of small-amplitude ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document}-limit cycles that can be made to bifurcate from the equilibrium when the parameters of the family are restricted to Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}. We obtain global upper bounds on the number of bifurcated ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document}-limit cycles.

An effective version of Schmüdgen’s Positivstellensatz for the hypercube

Let S subseteq mathbb {R}^n be a compact semialgebraic set and let f be a polynomial nonnegative on S. Schmüdgen’s Positivstellensatz then states that for any eta > 0, the nonnegativity of f + eta on S can be certified by expressing f + eta as a conic combination of products of the polynomials that occur in the inequalities defining S, where the coefficients are (globally nonnegative) sum-of-squares polynomials. It does not, however, provide explicit bounds on the degree of the polynomials required for such an expression. We show that in the special case where S = [-1, 1]^n is the hypercube, a Schmüdgen-type certificate of nonnegativity exists involving only polynomials of degree O(1 / sqrt{eta }). This improves quadratically upon the previously best known estimate in O(1/eta ). Our proof relies on an application of the polynomial kernel method, making use in particular of the Jackson kernel on the interval [-1, 1].

Exponential convexifying of polynomials

Let X⊂Rn be a convex closed and semialgebraic set and let f be a polynomial positive on X. We prove that there exists an exponent N⩾1, such that for any ξ∈Rn the function φN(x)=eN|x−ξ|2f(x) is strongly convex on X. When X is unbounded we have to assume also that the leading form of f is positive in Rn∖{0}. We obtain strong convexity of ΦN(x)=eeN|x|2f(x) on possibly unbounded X, provided N is sufficiently large, assuming only that f is positive on X. We apply these results for searching critical points of polynomials on convex closed semialgebraic sets.

On the effective Putinar’s Positivstellensatz and moment approximation

We analyse the representation of positive polynomials in terms of Sums of Squares. We provide a quantitative version of Putinar’s Positivstellensatz over a compact basic semialgebraic set S, with a new polynomial bound on the degree of the positivity certificates. This bound involves a Łojasiewicz exponent associated to the description of S. We show that if the gradients of the active constraints are linearly independent on S (Constraint Qualification condition), this Łojasiewicz exponent is equal to 1. We deduce the first general polynomial bound on the convergence rate of the optima in Lasserre’s Sum-of-Squares hierarchy to the global optimum of a polynomial function on S, and the first general bound on the Hausdorff distance between the cone of truncated (probability) measures supported on S and the cone of truncated pseudo-moment sequences, which are positive on the quadratic module of S.

The Algebraic Boundary of the Sonc-Cone

In this article, we explore the connections between nonnegativity, the theory of $A$-discriminants, and tropical geometry. For an integral support set $A \subset \mathbb{Z}^n$, we cover the boundary of the sonc-cone by semialgebraic sets that are parametrized by families of tropical hypersurfaces. As an application, we give sufficient conditions for equality between the sonc-cone and the sparse nonnegativity cone for generic support sets, and we describe a semialgebraic stratification of the boundary of the sonc-cone in the univariate case.

Transcendental Properties of Entropy-Constrained Sets

For information-theoretic quantities with an asymptotic operational characterization, the question arises whether an alternative single-shot characterization exists, possibly including an optimization over an ancilla system. If the expressions are algebraic and the ancilla is finite, this leads to semialgebraic level sets. In this work, we provide a criterion for disproving that a set is semialgebraic based on an analytic continuation of the Gauss map. Applied to the von Neumann entropy, this shows that its level sets are nowhere semialgebraic in dimension dge 3, ruling out algebraic single-shot characterizations with finite ancilla (e.g., via catalytic transformations). We show similar results for related quantities, including the relative entropy, and discuss under which conditions entropy values are transcendental, algebraic, or rational.

Positive dimensional parametric polynomial systems, connectivity queries and applications in robotics

In this paper we introduce methods and algorithms that will help us solve connectivity queries of parameterized semi-algebraic sets. Answering these connectivity queries is applied in the design of robotic structures having similar kinematic properties (e.g. topology of the kinematic-singularity-free space). From these algorithms one also obtain solutions to connectivity queries of a specific parameter which is in turn related to kinematic-singularity free path-planning of a specific manipulator belonging to the family of robots with these properties; i.e. we obtain paths joining two given singularity free configurations lying in the same connected component of the singularity-free space.We prove in the paper how one reduces the problems related to connectivity queries of parameterized semi-algebraic sets to closed and bounded semi-algebraic sets. We then design an algorithm using computer-algebra methods for “solving” positive dimensional polynomial system depending on parameters. The meaning of solving here means partitioning the parameter space into semi-algebraic components over which the number of connected components of the semi-algebraic set defined by the input system is invariant. The complexity of this algorithm is singly exponential in the dimension of the ambient space. The algorithm scales enough to analyze automatically the family of UR-series robots.Finally we provide manual analysis of the family of UR-series robots, proving that the number of connected components of the complementary of kinematic singularity set of a generic UR-robot is eight.

Families of faces and the normal cycle of a convex semi-algebraic set

We study families of faces for convex semi-algebraic sets via the normal cycle which is a semi-algebraic set similar to the conormal variety in projective duality theory. We propose a convex algebraic notion of a patch—a term recently coined by Ciripoi, Kaihnsa, Löhne, and Sturmfels as a tool for approximating the convex hull of a semi-algebraic set. We discuss geometric consequences, both for the semi-algebraic and convex geometry of the families of faces, as well as variations of our definition and their consequences.

Tangencies and polynomial optimization

Given a polynomial function $$f :{\mathbb {R}}^n \rightarrow {\mathbb {R}}$$ and an unbounded closed semi-algebraic set $$S \subset {\mathbb {R}}^n,$$ we show that the conditions listed below are characterized exactly in terms of the so-called tangency variety of the restriction of f on S: Besides, we also provide some stability criteria for boundedness and coercivity of f on S.

Existence of Pareto Solutions for Vector Polynomial Optimization Problems with Constraints

This paper deals with a vector polynomial optimization problem over a basic closed semi-algebraic set. By invoking some powerful tools from real semi-algebraic geometry, we first introduce the concept called tangency varieties; obtain the relationships of the Palais–Smale condition, Cerami condition, M-tameness, and properness related to the considered problem, in which the condition of Mangasarian–Fromovitz constraint qualification at infinity plays an essential role in deriving these relationships. At last, according to the obtained connections, we establish the existence of Pareto solutions to the problem in consideration and give some examples to illustrate our main findings.

$C^{m}$ semialgebraic sections over the plane

In this paper we settle the two-dimensional case of a conjecture involving unknown semialgebraic functions with specified smoothness. More precisely, we prove the following result: Let $\mathcal{H}$ be a semialgebraic bundle with respect to $C^m_{loc}(\mathbb{R}^2, \mathbb{R}^{D})$. If $\mathcal{H}$ has a section, then it has a semialgebraic section.

On the Reeb Spaces of Definable Maps

We prove that the Reeb space of a proper definable map $$f:X \rightarrow Y$$ in an arbitrary o-minimal expansion of a real closed field is realizable as a proper definable quotient. This result can be seen as an o-minimal analog of Stein factorization of proper morphisms in algebraic geometry. We also show that the Betti numbers of the Reeb space of f can be arbitrarily large compared to those of X, unlike in the special case of Reeb graphs of manifolds. Nevertheless, in the special case when $$f:X \rightarrow Y$$ is a semi-algebraic map and X is closed and bounded, we prove a singly exponential upper bound on the Betti numbers of the Reeb space of f in terms of the number and degrees of the polynomials defining X, Y, and f.

Chow's theorem for real analytic Levi-flat hypersurfaces

In this article we provide a version of Chow's theorem for real analytic Levi-flat hypersurfaces in the complex projective space Pn, n≥2. More specifically, we prove that a real analytic Levi-flat hypersurface M⊂Pn, with singular set of real dimension at most 2n−4 and whose Levi leaves are contained in algebraic hypersurfaces, is tangent to the levels of a rational function in Pn. As a consequence, M is a semialgebraic set. We also prove that a Levi foliation on Pn — a singular real analytic foliation whose leaves are immersed complex manifolds of codimension one — satisfying similar conditions — singular set of real dimension at most 2n−4 and all leaves algebraic — is defined by the level sets of a rational function.

A Sparse Version of Reznick’s Positivstellensatz

If f is a positive definite form, Reznick’s Positivstellensatz states that there exists [Formula: see text] such that [Formula: see text] is a sum of squares of polynomials. Assuming that f can be written as a sum of forms [Formula: see text], where each fl depends on a subset of the initial variables, and assuming that these subsets satisfy the so-called running intersection property, we provide a sparse version of Reznick’s Positivstellensatz. Namely, there exists [Formula: see text] such that [Formula: see text], where σl is a sum of squares of polynomials, Hl is a uniform polynomial denominator, and both polynomials [Formula: see text] involve the same variables as fl for each [Formula: see text]. In other words, the sparsity pattern of f is also reflected in this sparse version of Reznick’s certificate of positivity. We next use this result to also obtain positivity certificates for (i) polynomials nonnegative on the whole space and (ii) polynomials nonnegative on a (possibly noncompact) basic semialgebraic set, assuming that the input data satisfy the running intersection property. Both are sparse versions of a positivity certificate from Putinar and Vasilescu. Funding: V. Magron was supported by the Fondation Mathématique Jacques Hadamard Programme Gaspard Monge for Optimization (Exact Polynomial Optimization with Innovative Certifed Schemes project). This work has benefited from the Tremplin European Research Council Starting [Grant ANR-18-ERC2-0004-01] (Tremplin-Certified Optimization for Cyber-Physical Systems project), the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Actions [Grant 813211] (Polynomial Optimization, Efficiency through Moments and Algebra) as well as from the Artificial Intelligence Interdisciplinary Institute Artificial and Natural Intelligence Toulouse Institute funding, through the French “Investing for the Future Programme d’Investissements d’Avenir3” program [Grant n°ANR-19-PI3A-0004].