Semi-algebraic Sets Research Articles

Universality Theorems for Inscribed Polytopes and Delaunay Triangulations

We prove that every primary basic semi-algebraic set is homotopy equivalent to the set of inscribed realizations (up to Möbius transformation) of a polytope. If the semi-algebraic set is, moreover, open, it is, additionally, (up to homotopy) the retract of the realization space of some inscribed neighborly (and simplicial) polytope. We also show that all algebraic extensions of \({\mathbb {Q}}\) are needed to coordinatize inscribed polytopes. These statements show that inscribed polytopes exhibit the Mnëv universality phenomenon. Via stereographic projections, these theorems have a direct translation to universality theorems for Delaunay subdivisions. In particular, the realizability problem for Delaunay triangulations is polynomially equivalent to the existential theory of the reals.

Simultaneous resolution of singularities in the Nash category: finiteness and effectiveness

In this paper we present new proofs using real spectra of the finiteness theorem on Nash trivial simultaneous resolution and the finiteness theorem on Blow-Nash triviality for isolated real algebraic singularities. That is, we prove that a family of Nash sets in a Nash manifold indexed by a semialgebraic set always admits a Nash trivial simultaneous resolution after a partition of the parameter space into finitely many semialgebraic pieces and in the case of isolated singularities it admits a finite Blow-Nash trivialization. We also complement the finiteness results with recursive bounds.

On regularization of a certain class of distributions

We construct a regularization of a distribution of the form , where f is a tempered distribution and a function is infinitely differentiable outside a semialgebraic set N and has power singularities together with any its derivative.

Matrix convex hulls of free semialgebraic sets

This article resides in the realm of the noncommutative (free) analog of real algebraic geometry - the study of polynomial inequalities and equations over the real numbers - with a focus on matrix convex sets $C$ and their projections $\hat C$. A free semialgebraic set which is convex as well as bounded and open can be represented as the solution set of a Linear Matrix Inequality (LMI), a result which suggests that convex free semialgebraic sets are rare. Further, Tarski's transfer principle fails in the free setting: The projection of a free convex semialgebraic set need not be free semialgebraic. Both of these results, and the importance of convex approximations in the optimization community, provide impetus and motivation for the study of the free (matrix) convex hull of free semialgebraic sets. This article presents the construction of a sequence $C^{(d)}$ of LMI domains in increasingly many variables whose projections $\hat C^{(d)}$ are successively finer outer approximations of the matrix convex hull of a free semialgebraic set $D_p=\{X: p(X)\succeq0\}$. It is based on free analogs of moments and Hankel matrices. Such an approximation scheme is possibly the best that can be done in general. Indeed, natural noncommutative transcriptions of formulas for certain well known classical (commutative) convex hulls does not produce the convex hulls in the free case. This failure is illustrated on one of the simplest free nonconvex $D_p$. A basic question is which free sets $\hat S$ are the projection of a free semialgebraic set $S$? Techniques and results of this paper bear upon this question which is open even for convex sets.

CR-continuation of arc-analytic maps

Given a set $E$ in $\mathbb {C}^m$ and a point $p\in E$, there is a unique smallest complex-analytic germ $X_p$ containing $E_p$, called the holomorphic closure of $E_p$. We study the holomorphic closure of semialgebraic arc-symmetric sets. Our main application concerns CR-continuation of semialgebraic arc-analytic mappings: A mapping $f:M\to \mathbb {C}^n$ on a connected real-analytic CR manifold which is semialgebraic arc-analytic and CR on a non-empty open subset of $M$ is CR on the whole $M$.

Obstacle detection and avoidance with noisy measurements using danger zone concepts

Potential collision danger increases if a vehicle gradually moves closer to other vehicles or obstacles. If the collision time can be predicted before the vehicle is dangerously close to other vehicles or obstacles, the vehicle can devise an effective evasive maneuver or reroute its path, thus avoiding collision. This paper proposes an obstacle avoidance algorithm for unmanned vehicles in an unknown environment. The proposed avoidance algorithm uses the danger zone concept to determine whether the obstacle or surrounding vehicle will cause a possible collision. The danger zone is constructed around the vehicle by using semi-algebraic functions. The possible collision can be predicted by checking the sign of these semi-algebraic functions. The semi-algebraic functions for constructing the danger zone apply the relative velocities between the vehicle, and other vehicles or obstacles and can be used for both two- and three-dimensional cases. Several avoidance schemes are provided, with examples to demonstrate the concept of the proposed approach.

Remez-type inequality on sets with cusps

A Remez-type inequality is proved for a large family of sets with cusps in RN, including compact, fat and semialgebraic (or subanalytic) sets.

The structure of completely positive matrices according to their CP-rank and CP-plus-rank

We study the topological properties of the cp-rank operator cp(A) and the related cp-plus-rank operator cp+(A) (which is introduced in this paper) in the set Sn of symmetric n×n-matrices. For the set of completely positive matrices, CPn, we show that for any fixed p the set of matrices A satisfying cp(A)=cp+(A)=p is open in Sn∖bd(CPn). We also prove that the set An of matrices with cp(A)=cp+(A) is dense in Sn. By applying the theory of semi-algebraic sets we are able to show that membership in An is even a generic property. We furthermore answer several questions on the existence of matrices satisfying cp(A)=cp+(A) or cp(A)≠cp+(A), and establish genericity of having infinitely many minimal cp-decompositions.

Positive primitive formulae of modules over rings of semi-algebraic functions on a curve

Let R be a real closed field, and $${X\subseteq R^m}$$X⊆Rm semi-algebraic and 1-dimensional. We consider complete first-order theories of modules over the ring of continuous semi-algebraic functions $${X\to R}$$X?R definable with parameters in R. As a tool we introduce (pre)-piecewise vector bundles on X and show that the category of piecewise vector bundles on X is equivalent to the category of syzygies of finitely generated submodules of free modules. We give an explicit method to determine the Baur---Monk invariants of free modules in terms of pre-piecewise vector bundles. When R is a recursive real closed field this yields the decidability of the theory of free modules. Where it makes sense, we address the same questions for continuous definable functions in o-minimal expansions of a real closed field. From the free module case we are able to deduce generalisations of some results to arbitrary modules over the ring. We present a geometrically motivated quantifier elimination result down to the level of positive primitive formulae with a certain block decomposition of the matrix of coefficients.

Clarke Subgradients for Directionally Lipschitzian Stratifiable Functions

Using a geometric argument, we show that under a reasonable continuity condition, the Clarke subdifferential of a semi-algebraic (or more generally stratifiable) directionally Lipschitzian function admits a simple form: The normal cone to the domain and limits of gradients generate the entire Clarke subdifferential. The characterization formula we obtain unifies various apparently disparate results that have appeared in the literature. Our techniques also yield a simplified proof that closed semialgebraic functions on Rn have a limiting subdifferential graph of uniform local dimension n.

Positivstellensatz for semi-algebraic sets in real closed valued fields

Positivstellensatz for semi-algebraic sets in real closed valued fields

An Inertial Tseng’s Type Proximal Algorithm for Nonsmooth and Nonconvex Optimization Problems

We investigate the convergence of a forward---backward---forward proximal-type algorithm with inertial and memory effects when minimizing the sum of a nonsmooth function with a smooth one in the absence of convexity. The convergence is obtained provided an appropriate regularization of the objective satisfies the Kurdyka---źojasiewicz inequality, which is for instance fulfilled for semi-algebraic functions.

Computation of feedback control for time optimal state transfer using Groebner basis

Computation of time optimal feedback control law for a controllable linear time invariant system with bounded inputs is considered. Unlike a recent paper by the authors, the target final state is not limited to the origin of state-space, but is allowed to be in the set of constrained controllable states. Switching surfaces are formulated as semi-algebraic sets using Groebner basis based elimination theory. Using these semi-algebraic sets, a nested switching logic is synthesized to generate the time optimal feedback control. However, the optimal control law enforces an unavoidable limit cycle in the time-optimal trajectory for most non-origin target points. The time-period of this limit-cycle is dependent on the target position. This dependence is algebraically characterized and a method to compute the time-period of the limit-cycle is provided. As a natural extension, the set of constrained controllable states is also computed.

Algebraic boundaries of convex semi-algebraic sets

We study the algebraic boundary of a convex semi-algebraic set via duality in convex and algebraic geometry. We generalise the correspondence of facets of a polytope with the vertices of the dual polytope to general semi-algebraic convex sets. In this case, exceptional families of extreme points might exist and we characterise them semi-algebraically. We also give a strategy for computing a complete list of exceptional families, given the algebraic boundary of the dual convex set.

Determining “small parameters” for quasi-steady state

For a parameter-dependent system of ordinary differential equations we present a systematic approach to the determination of parameter values near which singular perturbation scenarios (in the sense of Tikhonov and Fenichel) arise. We call these special values Tikhonov–Fenichel parameter values. The principal application we intend is to equations that describe chemical reactions, in the context of quasi-steady state (or partial equilibrium) settings. Such equations have rational (or even polynomial) right-hand side. We determine the structure of the set of Tikhonov–Fenichel parameter values as a semi-algebraic set, and present an algorithmic approach to their explicit determination, using Groebner bases. Examples and applications (which include the irreversible and reversible Michaelis–Menten systems) illustrate that the approach is rather easy to implement.

Strong duality in Lasserre’s hierarchy for polynomial optimization

A polynomial optimization problem (POP) consists of minimizing a multivariate real polynomial on a semi-algebraic set \(K\) described by polynomial inequalities and equations. In its full generality it is a non-convex, multi-extremal, difficult global optimization problem. More than an decade ago, J. B. Lasserre proposed to solve POPs by a hierarchy of convex semidefinite programming (SDP) relaxations of increasing size. Each problem in the hierarchy has a primal SDP formulation (a relaxation of a moment problem) and a dual SDP formulation (a sum-of-squares representation of a polynomial Lagrangian of the POP). In this note, we show that there is no duality gap between each primal and dual SDP problem in Lasserre’s hierarchy, provided one of the constraints in the description of set \(K\) is a ball constraint. Our proof uses elementary results on SDP duality, and it does not assume that \(K\) has a strictly feasible point.

A Note on Semialgebraically Proper Maps

We prove that a semialgebraic map is semialgebraically proper if and only if it is proper. As an application of this assertion, we compare the semialgebraically proper actions with proper actions in a sense of Palais.

Semidefinite Approximations of Projections and Polynomial Images of SemiAlgebraic Sets

Given a compact semialgebraic set $\mathbf{S} \subset \mathbb{R}^n$ and a polynomial map $\mathbf{f} : \mathbb{R}^n \to \mathbb{R}^m$, we consider the problem of approximating the image set $\mathbf{F}=\mathbf{f}(\mathbf{S}) \subset \mathbb{R}^m$. This includes in particular the projection of $\mathbf{S}$ on $\mathbb{R}^m$ for $n \geq m$. Assuming that $\mathbf{F} \subset \mathbf{B}$, with $\mathbf{B} \subset \mathbb{R}^m$ being a “simple” set (e.g., a box or a ball), we provide two methods to compute certified outer approximations of $\mathbf{F}$. Method 1 exploits the fact that $\mathbf{F}$ can be defined with an existential quantifier, while Method 2 computes approximations of the support of image measures. The two methods output a sequence of superlevel sets defined with a single polynomial that yield explicit outer approximations of $\mathbf{F}$. Finding the coefficients of this polynomial boils down to computing an optimal solution of a convex semidefinite program. We provide guarantees of strong convergence to $\mathbf{F}$ in $L_1$ norm on $\mathbf{B}$, when the degree of the polynomial approximation tends to infinity. Several examples of applications are provided, together with numerical experiments.

Semidefinite Representations of Noncompact Convex Sets

We consider the problem of the semidefinite representation of a class of noncompact basic semialgebraic sets. We introduce the conditions of pointedness and closedness at infinity of a semialgebraic set and show that under these conditions our modified hierarchies of nested theta bodies and Lasserre's relaxations converge to the closure of the convex hull of $S$. Moreover, if the Putinar-Prestel's Bounded Degree Representation (PP-BDR) property is satisfied, our theta body and Lasserre's relaxation are exact when the order is large enough; if the PP-BDR property does not hold, our hierarchies converge uniformly to the closure of the convex hull of $S$ restricted to every fixed ball centered at the origin. We illustrate through a set of examples that the conditions of pointedness and closedness are essential to ensure the convergence. Finally, we provide some strategies to deal with cases where the conditions of pointedness and closedness are violated.

Semidefinite Programming For Chance Constrained Optimization Over Semialgebraic Sets

In this paper, “chance optimization” problems are introduced, where one aims at maximizing the probability of a set defined by polynomial inequalities. These problems are, in general, nonconvex and computationally hard. With the objective of developing systematic numerical procedures to solve such problems, a sequence of convex relaxations based on the theory of measures and moments is provided, whose sequence of optimal values is shown to converge to the optimal value of the original problem. Indeed, we provide a sequence of semidefinite programs of increasing dimension which can arbitrarily approximate the solution of the original problem. To be able to efficiently solve the resulting large-scale semidefinite relaxations, a first-order augmented Lagrangian algorithm is implemented. Numerical examples are presented to illustrate the computational performance of the proposed approach.