1,829 research outputs found
Certification of Real Inequalities -- Templates and Sums of Squares
We consider the problem of certifying lower bounds for real-valued
multivariate transcendental functions. The functions we are dealing with are
nonlinear and involve semialgebraic operations as well as some transcendental
functions like , , , etc. Our general framework is to use
different approximation methods to relax the original problem into polynomial
optimization problems, which we solve by sparse sums of squares relaxations. In
particular, we combine the ideas of the maxplus estimators (originally
introduced in optimal control) and of the linear templates (originally
introduced in static analysis by abstract interpretation). The nonlinear
templates control the complexity of the semialgebraic relaxations at the price
of coarsening the maxplus approximations. In that way, we arrive at a new -
template based - certified global optimization method, which exploits both the
precision of sums of squares relaxations and the scalability of abstraction
methods. We analyze the performance of the method on problems from the global
optimization literature, as well as medium-size inequalities issued from the
Flyspeck project.Comment: 27 pages, 3 figures, 4 table
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
Lower Bounds on Complexity of Lyapunov Functions for Switched Linear Systems
We show that for any positive integer , there are families of switched
linear systems---in fixed dimension and defined by two matrices only---that are
stable under arbitrary switching but do not admit (i) a polynomial Lyapunov
function of degree , or (ii) a polytopic Lyapunov function with facets, or (iii) a piecewise quadratic Lyapunov function with
pieces. This implies that there cannot be an upper bound on the size of the
linear and semidefinite programs that search for such stability certificates.
Several constructive and non-constructive arguments are presented which connect
our problem to known (and rather classical) results in the literature regarding
the finiteness conjecture, undecidability, and non-algebraicity of the joint
spectral radius. In particular, we show that existence of an extremal piecewise
algebraic Lyapunov function implies the finiteness property of the optimal
product, generalizing a result of Lagarias and Wang. As a corollary, we prove
that the finiteness property holds for sets of matrices with an extremal
Lyapunov function belonging to some of the most popular function classes in
controls
Certification of inequalities involving transcendental functions: combining SDP and max-plus approximation
We consider the problem of certifying an inequality of the form ,
, where is a multivariate transcendental function, and
is a compact semialgebraic set. We introduce a certification method, combining
semialgebraic optimization and max-plus approximation. We assume that is
given by a syntaxic tree, the constituents of which involve semialgebraic
operations as well as some transcendental functions like , ,
, etc. We bound some of these constituents by suprema or infima of
quadratic forms (max-plus approximation method, initially introduced in optimal
control), leading to semialgebraic optimization problems which we solve by
semidefinite relaxations. The max-plus approximation is iteratively refined and
combined with branch and bound techniques to reduce the relaxation gap.
Illustrative examples of application of this algorithm are provided, explaining
how we solved tight inequalities issued from the Flyspeck project (one of the
main purposes of which is to certify numerical inequalities used in the proof
of the Kepler conjecture by Thomas Hales).Comment: 7 pages, 3 figures, 3 tables, Appears in the Proceedings of the
European Control Conference ECC'13, July 17-19, 2013, Zurich, pp. 2244--2250,
copyright EUCA 201
Polynomial Bounds for Oscillation of Solutions of Fuchsian Systems
We study the problem of placing effective upper bounds for the number of
zeros of solutions of Fuchsian systems on the Riemann sphere. The principal
result is an explicit (non-uniform) upper bound, polynomially growing on the
frontier of the class of Fuchsian systems of given dimension n having m
singular points. As a function of n,m, this bound turns out to be double
exponential in the precise sense explained in the paper. As a corollary, we
obtain a solution of the so called restricted infinitesimal Hilbert 16th
problem, an explicit upper bound for the number of isolated zeros of Abelian
integrals which is polynomially growing as the Hamiltonian tends to the
degeneracy locus. This improves the exponential bounds recently established by
A. Glutsyuk and Yu. Ilyashenko.Comment: Will appear in Annales de l'institut Fourier vol. 60 (2010
Certification of Bounds of Non-linear Functions: the Templates Method
The aim of this work is to certify lower bounds for real-valued multivariate
functions, defined by semialgebraic or transcendental expressions. The
certificate must be, eventually, formally provable in a proof system such as
Coq. The application range for such a tool is widespread; for instance Hales'
proof of Kepler's conjecture yields thousands of inequalities. We introduce an
approximation algorithm, which combines ideas of the max-plus basis method (in
optimal control) and of the linear templates method developed by Manna et al.
(in static analysis). This algorithm consists in bounding some of the
constituents of the function by suprema of quadratic forms with a well chosen
curvature. This leads to semialgebraic optimization problems, solved by
sum-of-squares relaxations. Templates limit the blow up of these relaxations at
the price of coarsening the approximation. We illustrate the efficiency of our
framework with various examples from the literature and discuss the interfacing
with Coq.Comment: 16 pages, 3 figures, 2 table
Computing the homology of basic semialgebraic sets in weak exponential time
We describe and analyze an algorithm for computing the homology (Betti
numbers and torsion coefficients) of basic semialgebraic sets which works in
weak exponential time. That is, out of a set of exponentially small measure in
the space of data the cost of the algorithm is exponential in the size of the
data. All algorithms previously proposed for this problem have a complexity
which is doubly exponential (and this is so for almost all data)
Grothendieck ring of semialgebraic formulas and motivic real Milnor fibres
We define a Grothendieck ring for basic real semialgebraic formulas, that is
for systems of real algebraic equations and inequalities. In this ring the
class of a formula takes into consideration the algebraic nature of the set of
points satisfying this formula and contains as a ring the usual Grothendieck
ring of real algebraic formulas. We give a realization of our ring that allows
to express a class as a Z[1/2]- linear combination of classes of real algebraic
formulas, so this realization gives rise to a notion of virtual Poincar\'e
polynomial for basic semialgebraic formulas. We then define zeta functions with
coefficients in our ring, built on semialgebraic formulas in arc spaces. We
show that they are rational and relate them to the topology of real Milnor
fibres.Comment: 30 pages, 1 figur
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