112 research outputs found
The Moment Problem for Continuous Positive Semidefinite Linear functionals
Let be a locally convex topology on the countable dimensional
polynomial -algebra \rx:=\reals[X_1,...,X_n]. Let be a closed
subset of , and let be a finitely generated
quadratic module in \rx. We investigate the following question: When is the
cone \Pos(K) (of polynomials nonnegative on ) included in the closure of
? We give an interpretation of this inclusion with respect to representing
continuous linear functionals by measures. We discuss several examples; we
compute the closure of M=\sos with respect to weighted norm- topologies.
We show that this closure coincides with the cone \Pos(K) where is a
certain convex compact polyhedron.Comment: 14 page
A unified approach to computing real and complex zeros of zero-dimensional ideals
In this paper we propose a unified methodology for computing the set of complex () or real () roots of an ideal in , assuming is finite. We show how moment matrices, defined in terms of a given set of generators of the ideal I, can be used to (numerically) find not only the real variety , as shown in the authorsâ previous work, but also the complex variety , thus leading to a unified treatment of the algebraic and real algebraic problem. In contrast to the real algebraic version of the algorithm, the complex analogue only uses basic numerical linear algebra because it does not require positive semidefiniteness of the moment matrix and so avoids semidefinite programming techniques. The links between these algorithms and other numerical algebraic methods are outlined and their stopping criteria are related
Finding largest small polygons with GloptiPoly
A small polygon is a convex polygon of unit diameter. We are interested in
small polygons which have the largest area for a given number of vertices .
Many instances are already solved in the literature, namely for all odd ,
and for and 8. Thus, for even , instances of this problem
remain open. Finding those largest small polygons can be formulated as
nonconvex quadratic programming problems which can challenge state-of-the-art
global optimization algorithms. We show that a recently developed technique for
global polynomial optimization, based on a semidefinite programming approach to
the generalized problem of moments and implemented in the public-domain Matlab
package GloptiPoly, can successfully find largest small polygons for and
. Therefore this significantly improves existing results in the domain.
When coupled with accurate convex conic solvers, GloptiPoly can provide
numerical guarantees of global optimality, as well as rigorous guarantees
relying on interval arithmetic
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
An Optimization Approach to Weak Approximation of LĂ©vy-Driven Stochastic Differential Equations
We propose an optimization approach to weak approximation of LĂ©vy-driven stochastic differential equations. We employ a mathematical programming framework to obtain numerically upper and lower bound estimates of the target expectation, where the optimization procedure ends up with a polynomial programming problem. An advantage of our approach is that all we need is a closed form of the LĂ©vy measure, not the exact simulation knowledge of the increments or of a shot noise representation for the time discretization approximation. We also investigate methods for approximation at some different intermediate time points simultaneously
Somatosensory profiling of patients undergoing alcohol withdrawal: Do neuropathic pain and sensory loss represent a problem?
Chronic heavy alcohol use is known to cause neurological complications such as peripheral neuropathy. Concerning the pathophysiology, few sural nerve and skin biopsy studies showed that small fibers might be selectively vulnerable to degeneration in alcohol-related peripheral neuropathy. Pain has rarely been properly evaluated in this pathology. The present study aims at assessing pain intensity, potential neuropathic characteristics as well as the functionality of both small and large nerve sensitive fibers.
In this observational study, 27 consecutive adult patients, hospitalized for alcohol withdrawal and 13 healthy controls were recruited. All the participants underwent a quantitative sensory testing (QST) according to the standardized protocol of the German Research Network Neuropathic Pain, a neurological examination and filled standardized questionnaires assessing alcohol consumption and dependence as well as pain characteristics and psychological comorbidities.
Nearly half of the patients (13/27) reported pain. Yet, pain intensity was weak, leading to a low interference with daily life, and its characteristics did not support a neuropathic component. A functional impairment of small nerve fibers was frequently described, with thermal hypoesthesia observed in 52% of patients. Patients with a higher alcohol consumption over the last 2 years showed a greater impairment of small fiber function.
Patients report pain but it is however unlikely to be caused by peripheral neuropathy given the non-length-dependent distribution and the absence of neuropathic pain features. Chronic pain in AUD deserves to be better evaluated and managed as it represents an opportunity to improve long-term clinical outcomes, potentially participating to relapse prevention
The matricial relaxation of a linear matrix inequality
Given linear matrix inequalities (LMIs) L_1 and L_2, it is natural to ask:
(Q1) when does one dominate the other, that is, does L_1(X) PsD imply L_2(X)
PsD? (Q2) when do they have the same solution set? Such questions can be
NP-hard. This paper describes a natural relaxation of an LMI, based on
substituting matrices for the variables x_j. With this relaxation, the
domination questions (Q1) and (Q2) have elegant answers, indeed reduce to
constructible semidefinite programs. Assume there is an X such that L_1(X) and
L_2(X) are both PD, and suppose the positivity domain of L_1 is bounded. For
our "matrix variable" relaxation a positive answer to (Q1) is equivalent to the
existence of matrices V_j such that L_2(x)=V_1^* L_1(x) V_1 + ... + V_k^*
L_1(x) V_k. As for (Q2) we show that, up to redundancy, L_1 and L_2 are
unitarily equivalent.
Such algebraic certificates are typically called Positivstellensaetze and the
above are examples of such for linear polynomials. The paper goes on to derive
a cleaner and more powerful Putinar-type Positivstellensatz for polynomials
positive on a bounded set of the form {X | L(X) PsD}.
An observation at the core of the paper is that the relaxed LMI domination
problem is equivalent to a classical problem. Namely, the problem of
determining if a linear map from a subspace of matrices to a matrix algebra is
"completely positive".Comment: v1: 34 pages, v2: 41 pages; supplementary material is available in
the source file, or see http://srag.fmf.uni-lj.si
A convex polynomial that is not sos-convex
A multivariate polynomial is sos-convex if its Hessian
can be factored as with a possibly nonsquare
polynomial matrix . It is easy to see that sos-convexity is a sufficient
condition for convexity of . Moreover, the problem of deciding
sos-convexity of a polynomial can be cast as the feasibility of a semidefinite
program, which can be solved efficiently. Motivated by this computational
tractability, it has been recently speculated whether sos-convexity is also a
necessary condition for convexity of polynomials. In this paper, we give a
negative answer to this question by presenting an explicit example of a
trivariate homogeneous polynomial of degree eight that is convex but not
sos-convex. Interestingly, our example is found with software using sum of
squares programming techniques and the duality theory of semidefinite
optimization. As a byproduct of our numerical procedure, we obtain a simple
method for searching over a restricted family of nonnegative polynomials that
are not sums of squares.Comment: 15 page
NP-hardness of Deciding Convexity of Quartic Polynomials and Related Problems
We show that unless P=NP, there exists no polynomial time (or even
pseudo-polynomial time) algorithm that can decide whether a multivariate
polynomial of degree four (or higher even degree) is globally convex. This
solves a problem that has been open since 1992 when N. Z. Shor asked for the
complexity of deciding convexity for quartic polynomials. We also prove that
deciding strict convexity, strong convexity, quasiconvexity, and
pseudoconvexity of polynomials of even degree four or higher is strongly
NP-hard. By contrast, we show that quasiconvexity and pseudoconvexity of odd
degree polynomials can be decided in polynomial time.Comment: 20 page
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