23,742 research outputs found
Semi-definite programming and functional inequalities for Distributed Parameter Systems
We study one-dimensional integral inequalities, with quadratic integrands, on
bounded domains. Conditions for these inequalities to hold are formulated in
terms of function matrix inequalities which must hold in the domain of
integration. For the case of polynomial function matrices, sufficient
conditions for positivity of the matrix inequality and, therefore, for the
integral inequalities are cast as semi-definite programs. The inequalities are
used to study stability of linear partial differential equations.Comment: 8 pages, 5 figure
Infeasibility certificates for linear matrix inequalities
Farkas' lemma is a fundamental result from linear programming providing
linear certificates for infeasibility of systems of linear inequalities. In
semidefinite programming, such linear certificates only exist for strongly
infeasible linear matrix inequalities. We provide nonlinear algebraic
certificates for all infeasible linear matrix inequalities in the spirit of
real algebraic geometry. More precisely, we show that a linear matrix
inequality is infeasible if and only if -1 lies in the quadratic module
associated to it. We prove exponential degree bounds for the corresponding
algebraic certificate. In order to get a polynomial size certificate, we use a
more involved algebraic certificate motivated by the real radical and Prestel's
theory of semiorderings. Completely different methods, namely complete
positivity from operator algebras, are employed to consider linear matrix
inequality domination.Comment: 30 page
Bounded real lemmas for positive descriptor systems
A well known result in the theory of linear positive systems is the existence of positive definite diagonal matrix (PDDM) solutions to some well known linear matrix inequalities (LMIs). In this paper, based on the positivity characterization, a novel bounded real lemma for continuous positive descriptor systems in terms of strict LMI is first established by the separating hyperplane theorem. The result developed here provides a necessary and sufficient condition for systems to possess H?H? norm less than ? and shows the existence of PDDM solution. Moreover, under certain condition, a simple model reduction method is introduced, which can preserve positivity, stability and H?H? norm of the original systems. An advantage of such method is that systems? matrices of the reduced order systems do not involve solving of LMIs conditions. Then, the obtained results are extended to discrete case. Finally, a numerical example is given to illustrate the effectiveness of the obtained results
Matrix positivity preservers in fixed dimension. I
A classical theorem proved in 1942 by I.J. Schoenberg describes all
real-valued functions that preserve positivity when applied entrywise to
positive semidefinite matrices of arbitrary size; such functions are
necessarily analytic with non-negative Taylor coefficients. Despite the great
deal of interest generated by this theorem, a characterization of functions
preserving positivity for matrices of fixed dimension is not known.
In this paper, we provide a complete description of polynomials of degree
that preserve positivity when applied entrywise to matrices of dimension .
This is the key step for us then to obtain negative lower bounds on the
coefficients of analytic functions so that these functions preserve positivity
in a prescribed dimension. The proof of the main technical inequality is
representation theoretic, and employs the theory of Schur polynomials.
Interpreted in the context of linear pencils of matrices, our main results
provide a closed-form expression for the lowest critical value, revealing at
the same time an unexpected spectral discontinuity phenomenon.
Tight linear matrix inequalities for Hadamard powers of matrices and a sharp
asymptotic bound for the matrix-cube problem involving Hadamard powers are
obtained as applications. Positivity preservers are also naturally interpreted
as solutions of a variational inequality involving generalized Rayleigh
quotients. This optimization approach leads to a novel description of the
simultaneous kernels of Hadamard powers, and a family of stratifications of the
cone of positive semidefinite matrices.Comment: Changed notation for extreme critical value from to
. Addressed referee remarks to improve exposition, including
Remarks 1.2 and 3.3. Final version, 39 pages, to appear in Advances in
Mathematic
PIETOOLS: A Matlab Toolbox for Manipulation and Optimization of Partial Integral Operators
In this paper, we present PIETOOLS, a MATLAB toolbox for the construction and
handling of Partial Integral (PI) operators. The toolbox introduces a new class
of MATLAB object, opvar, for which standard MATLAB matrix operation syntax
(e.g. +, *, ' e tc.) is defined. PI operators are a generalization of bounded
linear operators on infinite-dimensional spaces that form a *-subalgebra with
two binary operations (addition and composition) on the space RxL2. These
operators frequently appear in analysis and control of infinite-dimensional
systems such as Partial Differential equations (PDE) and Time-delay systems
(TDS). Furthermore, PIETOOLS can: declare opvar decision variables, add
operator positivity constraints, declare an objective function, and solve the
resulting optimization problem using a syntax similar to the sdpvar class in
YALMIP. Use of the resulting Linear Operator Inequalities (LOIs) are
demonstrated on several examples, including stability analysis of a PDE,
bounding operator norms, and verifying integral inequalities. The result is
that PIETOOLS, packaged with SOSTOOLS and MULTIPOLY, offers a scalable,
user-friendly and computationally efficient toolbox for parsing, performing
algebraic operations, setting up and solving convex optimization problems on PI
operators
On the positivity of polynomials on the complex unit disc via LMIs
Investigating positivity of polynomials over the complex unit disc is a relevant problem in electrical and computer engineering. This paper provides two sufficient and necessary conditions for solving this problem via linear matrix inequalities (LMIs). These conditions are obtained by exploiting trigonometric transformations, a key tool for the representation of polynomials, and results from the theory of positive polynomials. Some numerical examples illustrate the proposed conditions. © 2012 IEEE.published_or_final_versio
Positivity, entanglement entropy, and minimal surfaces
The path integral representation for the Renyi entanglement entropies of
integer index n implies these information measures define operator correlation
functions in QFT. We analyze whether the limit , corresponding
to the entanglement entropy, can also be represented in terms of a path
integral with insertions on the region's boundary, at first order in .
This conjecture has been used in the literature in several occasions, and
specially in an attempt to prove the Ryu-Takayanagi holographic entanglement
entropy formula. We show it leads to conditional positivity of the entropy
correlation matrices, which is equivalent to an infinite series of polynomial
inequalities for the entropies in QFT or the areas of minimal surfaces
representing the entanglement entropy in the AdS-CFT context. We check these
inequalities in several examples. No counterexample is found in the few known
exact results for the entanglement entropy in QFT. The inequalities are also
remarkable satisfied for several classes of minimal surfaces but we find
counterexamples corresponding to more complicated geometries. We develop some
analytic tools to test the inequalities, and as a byproduct, we show that
positivity for the correlation functions is a local property when supplemented
with analyticity. We also review general aspects of positivity for large N
theories and Wilson loops in AdS-CFT.Comment: 36 pages, 10 figures. Changes in presentation and discussion of
Wilson loops. Conclusions regarding entanglement entropy unchange
Disentangling positivity constraints for generalized parton distributions
Positivity constraints are derived for the generalized parton distributions
(GPDs) of spin-1/2 hadrons. The analysis covers the full set of eight twist-2
GPDs. Several new inequalities are obtained which constrain GPDs by various
combinations of usual (forward) unpolarized and polarized parton distributions
including the transversity distribution.Comment: 9 pages (REVTEX), typos correcte
Inequalities for nucleon generalized parton distributions with helicity flip
Several positivity bounds are derived for generalized parton distributions
(GPDs) with helicity flip.Comment: 20 page
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