chaos in the hamiltonian mean field model rst show that the largest lyapunov exponent remains strictly positive in the innite size limit converging to its asymptotic value with 1 lnn corrections we then elucidate the scaling laws ruling the behavior of this asymptotic value in the critical region separating the ordered clustered phase and the disordered phase present at high energy densities we also show that the full spectrum of lyapunov exponents consists of a bulk component converging to the zero value taken by a test oscillator forced by the mean eld plus subextensive bands of o lnn exponents taking nite values we nally investigate the robustness of these results by studying a 2d extension of the hmf model where each particle is endowed with 4 degrees of freedom thus allowing the emergence of chaos at the level of single particle altogether these results illustrate the subtle eects of global or long range coupling and the importance of the order in which the innite time and innite size limits are taken for an innite size hmf system represented by the vlasov equation no chaos is present while chaos exists and subsists for any nite system size

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

quadratic maximization and semidefinite relaxation in this paper we study a class of quadratic maximization problems and their semidefinite programming sdp relaxation for a special subclass of the problems we show that the sdp relaxation provides an exact optimal solution another subclass which is hard guarantees that the sdp relaxation yields an approximate solution with a worst case performance ratio of 0 87856 this is a generalization of the well known result of goemans and williamson for the maximum cut problem finally we discuss extensions of these results in the presence of a certain type of sign restrictions

a projector based convergence proof of the ginelli algorithm for covariant lyapunov vectors abstract linear perturbations of solutions of dynamical systems exhibit different asymptotic growth rates which are naturally characterized by so called covariant lyapunov vectors clvs due to an increased interest of clvs in applications several algorithms were developed to compute them the ginelli algorithm is among the most commonly used although several properties of the algorithm have been analyzed there exists no mathematically rigorous convergence proof yet in this article we extend existing approaches in order to construct a projector based convergence proof of ginelli s algorithm one of the main ingredients will be an asymptotic characterization of clvs via the multiplicative ergodic theorem in the proof we keep a rather general setting allowing even for degenerate lyapunov spectra

a tensor product matrix approximation problem in quantum physics we consider a matrix approximation problem arising in the study of entanglement in quantum physics this notion represents a certain type of correlations between subsystems in a composite quantum system the states of a system are described by a density matrix which is a positive semidefinite matrix with trace one the goal is to approximate such a given density matrix by a so called separable density matrix and the distance between these matrices gives information about the degree of entanglement in the system separability here is expressed in terms of tensor products we discuss this approximation problem for a composite system with two subsystems and show that it can be written as a convex optimization problem with special structure we investigate related convex sets and suggest an algorithm for this approximation problem which exploits the tensor product structure in certain subproblems finally some computational results and experiences are presented

linear equations modulo 2 and the l_1 diameter of convex bodies we design a randomized polynomial time algorithm which given a 3 tensor of real numbers a a ijk i j k 1 n such that for all i j k in 1 dots n we have a ijk a ikj a kji a jik a kij a jki and a iik a ijj a iji 0 computes a number operatorname alg a which satisfies with probability at least frac12 omega sqrt frac log n n t cdot max x in 1 1 n sum i j k 1 n a ijk x ix jx k le operatorname alg a le max x in 1 1 n sum i j k 1 n a ijk x ix jx k on the other hand we show via a simple reduction from a result of hastad and venkatesh random structures algorithms 25 2004 pp 117 149 that under the assumption np not subseteq dtime n log n o 1 for every epsilon 0 there is no algorithm that approximates max x in 1 1 n sum i j k 1 n a ijk x ix jx k within a factor of 2 log n 1 epsilon in time 2 log n o 1 our algorithm is based on a reduction to the problem of computing the diameter of a convex body in mathbb r n with respect to the l 1 norm we show that it is possible to do so up to a multiplicative error of o sqrt frac n log n while no randomized polynomial time algorithm can achieve accuracy o sqrt frac n log n this resolves a question posed by brieden et al in mathematika 48 2001 pp 63 105 we apply our new algorithm to improve the algorithm of hastad and venkatesh for the max e3 lin 2 problem given an overdetermined system mathcal e of n linear equations modulo 2 in n le n boolean variables such that in each equation only three distinct variables appear the goal is to approximate in polynomial time the maximum number of satisfiable equations in mathcal e minus frac n 2 i e we subtract the expected number of satisfied equations in a random assignment hastad and venkatesh obtained an algorithm which approximates this value up to a factor of o sqrt n we obtain an o sqrt frac n log n approximation algorithm by relating this problem to the refutation problem for random 3 cnf formulas we give evidence that obtaining a significant improvement over this approximation factor is likely to be difficult

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

inhomogeneous polynomial optimization over a convex set an approximation approach in this paper we consider computational methods for optimizing a multivariate inhomogeneous polynomial function over a general convex set the focus is on the design and analysis of polynomial time approximation algorithms the methods are able to deal with optimization models with polynomial objective functions in any fixed degrees in particular we first study the problem of maximizing an inhomogeneous polynomial over the euclidean ball a polynomial time approximation algorithm is proposed for this problem with an assured relative worst case performance ratio which is dependent only on the dimensions of the model the method and approximation ratio are then generalized to optimize an inhomogeneous polynomial over the intersection of a finite number of co centered ellipsoids furthermore the constraint set is extended to a general convex compact set specifically we propose a polynomial time approximation algorithm with a relative worst case performance ratio for polynomial optimization over some convex compact sets e g a polytope finally numerical results are reported revealing good practical performance of the proposed algorithms for solving some randomly generated instances

on cones of nonnegative quadratic functions we derive linear matrix inequality lmi characterizations and dual decomposition algorithms for certain matrix cones which are generated by a given set using generalized co positivity these matrix cones are in fact cones of nonconvex quadratic functions that are nonnegative on a certain domain as a domain we consider for instance the intersection of a upper level set of a quadratic function and a half plane consequently we arrive at a generalization of yakubovich s s procedure result although the primary concern of this paper is to characterize the matrix cones by lmis we show as an application of our results that optimizing a general quadratic function over the intersection of an ellipsoid and a half plane can be formulated as semidefinite programming sdp thus proving the polynomiality of this class of optimization problems which arise e g from the application of the trust region method for nonlinear programming other applications are in control theory and robust optimization

polynomials nonnegative on a grid and discrete optimization we characterize the real valued polynomials on r n that are nonnegative not necessarily strictly positive on a grid k of points of r n in terms of a weighted sum of squares whose degree is bounded and known in advance we also show that the mimimization of an arbitrary polynomial on k a discrete optimization problem reduces to a convex continuous optimization problem of fixed size the case of concave polynomials is also investigated the proof is based on a recent result of curto and fialkow on the k moment problem

from stein s unbiased risk estimates to the method of generalized cross validation this paper concerns the method of generalized cross validation gcv a promising way of choosing between linear estimates based on stein estimates and the associated unbiased risk estimates stein 1981 a new approach to gcv is developed many consistency results are obtained for the cross validated steinized estimates in the contexts of nearest neighbor nonparametric regression model selection ridge regression and smoothing splines moreover the associated stein s unbiased risk estimate is shown to be uniformly consistent in assessing the true loss not the risk consistency properties are examined as well when the sampling error is unknown finally we propose a variant of gcv to handle the case that the dimension of the raw data is known to be greater than that of their expected values

local likelihood estimation abstract a scatterplot smoother is applied to data of the form x 1 y 1 x 2 y 2 xn yn and uses local fitting to estimate the dependence of y on x a simple example is the running lines smoother which fits a least squares line to the y values falling in a window around each x value the value of the estimated function at x is given by the value of the least squares line at x a smoother generalizes the least squares line which assumes that the dependence of y on x is linear in this article we extend the idea of local fitting to likelihood based regression models one such application is to the class of generalized linear models nelder and wedderburn 1972 we enlarge this class by replacing the covariate form 0 x 1 with an unspecified smooth function s x this function is estimated from the data by a technique we call local likelihood estimation the method consists of maximum likelihood estimation for 0 and 1 applied in a window around each x value multiple covariates are incor

hyperbolic decoupling of tangent space and effective dimension of dissipative systems we show using covariant lyapunov vectors that the tangent space of spatially extended dissipative systems is split into two hyperbolically decoupled subspaces one comprising a finite number of frequently entangled physical modes which carry the physically relevant information of the trajectory and a residual set of strongly decaying spurious modes the decoupling of the physical and spurious subspaces is defined by the absence of tangencies between them and found to take place generally we find evidence in partial differential equations in one and two spatial dimensions and even in lattices of coupled maps or oscillators we conjecture that the physical modes may constitute a local linear description of the inertial manifold at any point in the global attractor

spatiotemporal dynamics of the covariant lyapunov vectors of chaotic convection we explore the spatiotemporal dynamics of the spectrum of covariant lyapunov vectors for chaotic rayleigh b enard convection we use the inverse participation ratio to quantify the amount of spatial localization of the covariant lyapunov vectors the covariant lyapunov vectors are found to be spatially localized at times when the instantaneous covariant lyapunov exponents are large the spatial localization of the lyapunov vectors often occurs near defect structures in the fluid flow field there is an overall trend of decreasing spatial localization of the lyapunov vectors with increasing index of the vector the spatial localization of the covariant lyapunov vectors with positive lyapunov exponents decreases an order of magnitude faster with increasing vector index than all of the remaining vectors that we have computed we find that a weighted covariant lyapunov vector is useful for the visualization and interpretation of the significant connections between the lyapunov vectors and the flow field patterns

a semidefinite relaxation scheme for multivariate quartic polynomial optimization with quadratic constraints we present a general semidefinite relaxation scheme for general n variate quartic polynomial optimization under homogeneous quadratic constraints unlike the existing sum of squares approach which relaxes the quartic optimization problems to a sequence of typically large linear semidefinite programs sdp our relaxation scheme leads to a possibly nonconvex quadratic optimization problem with linear constraints over the semidefinite matrix cone in mathbb r n times n it is shown that each alpha factor approximate solution of the relaxed quadratic sdp can be used to generate in randomized polynomial time an o alpha factor approximate solution for the original quartic optimization problem where the constant in o cdot depends only on problem dimension in the case where only one positive definite quadratic constraint is present in the quartic optimization problem we present a randomized polynomial time approximation algorithm which can provide a guaranteed relative approximation ratio of 1 o n 2

further results on the consistent directions of least squares estimators wu 1980 defined the consistent directions of the least squares estimator in a linear model as the linear combinations of parameter estimates that are asymptotically consistent for the polynomial regression model a characterization of the space of consistent directions s was obtained in terms of the convergence rates of the corresponding design sequence to its limit points by employing a more general and yet simpler approach we obtain here a similar result for any regression model with one independent variable and smooth regression function when f i x is an extended tchebycheff system the above characterization is further refined and the consistency region c is shown to be either the set of limit points of the design sequence or the whole real line

complex quadratic optimization and semidefinite programming in this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the hermitian complex form a special case of the problem that we study corresponds to the max 3 cut model used in a recent paper of goemans and williamson j comput system sci 68 2004 pp 442 470 we first develop a closed form formula to compute the probability of a complex valued normally distributed bivariate random vector to be in a given angular region this formula allows us to compute the expected value of a randomized with a specific rounding rule solution based on the optimal solution of the complex semidefinite programming relaxation problem in particular we present an m 2 1 cos frac 2 pi m 8 pi approximation algorithm and then study the limit of that model in which the problem remains np hard we show that if the objective is to maximize a positive semidefinite hermitian form then the randomization rounding procedure guarantees a worst case performance ratio of pi 4 approx 0 7854 which is better than the ratio of 2 pi approx 0 6366 for its counterpart in the real case due to nesterov furthermore if the objective matrix is real valued positive semidefinite with nonpositive off diagonal elements then the performance ratio improves to 0 9349

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

eigenvalues and invariants of tensors abstract a tensor is represented by a supermatrix under a co ordinate system in this paper we define e eigenvalues and e eigenvectors for tensors and supermatrices by the resultant theory we define the e characteristic polynomial of a tensor an e eigenvalue of a tensor is a root of the e characteristic polynomial in the regular case a complex number is an e eigenvalue if and only if it is a root of the e characteristic polynomial we convert the e characteristic polynomial of a tensor to a monic polynomial and show that the coefficients of that monic polynomial are invariants of that tensor i e they are invariant under co ordinate system changes we call them principal invariants of that tensor the maximum number of principal invariants of mth order n dimensional tensors is a function of m and n we denote it by d m n and show that d 1 n 1 d 2 n n d m 2 m for m 3 and d m n m n 1 m for m n 3 we also define the rank of a tensor all real eigenvectors associated with nonzero e eigenvalues are in a subspace with dimension equal to its rank

fast numerical test of hyperbolic chaos an effective numerical method for testing the hyperbolicity of chaotic dynamics is suggested the method employs ideas of algorithms for covariant lyapunov vectors but avoids their explicit computation the outcome is a distribution of a characteristic value which is bounded within the unit interval and whose zero indicates a tangency between expanding and contracting subspaces to perform the test one has to solve several copies of equations for infinitesimal perturbations whose number is equal to the sum of numbers of positive and zero lyapunov exponents since this number is normally much less than the full phase space dimension this method provides a fast and memory saving way for numerical hyperbolicity testing

on maximization of quadratic form over intersection of ellipsoids with common center we demonstrate that if a1 a m are symmetric positive semidefinite n n matrices with positive definite sum and a is an arbitrary symmetric n n matrix then the relative accuracy in terms of the optimal value of the semidefinite relaxation max x tr ax mid tr a ix le1 i 1 m x succeq0 eqno hbox sdp of the optimization program x tax to max mid x ta ix le 1 i 1 m eqno hbox p is not worse than 1 frac 1 2 ln 2m 2 it is shown that this bound is sharp in order as far as the dependence on m is concerned and that a feasible solution x to p with x tax ge frac opt hbox rm sdp 2 ln 2m 2 eqno can be found efficiently this somehow improves one of the results of nesterov 4 where bound similar to is established for the case when all ai are of rank 1

symmetry properties of orthogonal and covariant lyapunov vectors and their exponents lyapunov exponents are indicators for the chaotic properties of a classical dynamical system they are most naturally defined in terms of the time evolution of a set of so called covariant vectors co moving with the linearized flow in tangent space taking a simple spring pendulum and the henon heiles system as examples we demonstrate the consequences of symplectic symmetry and of time reversal invariance for such vectors and study the transformation between different parameterizations of the flow this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

semidefinite relaxation bounds for indefinite homogeneous quadratic optimization this paper studies the relationship between the optimal value of a homogeneous quadratic optimization problem and its semidefinite programming sdp relaxation we consider two quadratic optimization models 1 min x c x mid x a k x ge 1 k 0 1 ldots m x in mathbb f n and 2 max x c x mid x a k x le 1 k 0 1 ldots m x in mathbb f n where mathbb f is either the real field mathbb r or the complex field mathbb c and a k c are symmetric matrices for the minimization model 1 we prove that if the matrix c and all but one of the a k s are positive semidefinite then the ratio between the optimal value of 1 and its sdp relaxation is upper bounded by o m 2 when mathbb f mathbb r and by o m when mathbb f mathbb c moreover when two or more of the a k s are indefinite this ratio can be arbitrarily large for the maximization model 2 we show that if c and at most one of the a k s are indefinite while other a k s are positive semidefinite then the ratio between the optimal value of 2 and its sdp relaxation is bounded from below by o 1 log m for both the real and the complex case this result improves the bound based on the so called approximate s lemma of ben tal nemirovski and roos siam j optim 13 2002 pp 535 560 when two or more of the a k s in 2 are indefinite we derive a general bound in terms of the problem data and the sdp solution for both optimization models we present examples to show that the derived approximation bounds are essentially tight

deterministic approximation algorithms for sphere constrained homogeneous polynomial optimization problems due to their fundamental nature and numerous applications sphere constrained polynomial optimization problems have received a lot of attention lately in this paper we consider three such problems i maximizing a homogeneous polynomial over the sphere ii maximizing a multilinear form over a cartesian product of spheres and iii maximizing a multiquadratic form over a cartesian product of spheres since these problems are generally intractable our focus is on designing polynomial time approximation algorithms for them by reducing the above problems to that of determining the l 2 diameters of certain convex bodies we show that they can all be approximated to within a factor of log n n d 2 1 deterministically where n is the number of variables and d is the degree of the polynomial this improves upon the currently best known approximation bound of 1 n d 2 1 in the literature we believe that our approach will find further applications in the design of approximation algorithms for polynomial optimization problems with provable guarantees

approximating quadratic programming with bound and quadratic constraints we consider the problem of approximating the global maximum of a quadratic program qp subject to bound and simple quadratic constraints based on several early results we show that a 4 7 approximate solution can be obtained in polynomial time

approximation bounds for quadratic optimization with homogeneous quadratic constraints we consider the np hard problem of finding a minimum norm vector in n dimensional real or complex euclidean space subject to m concave homogeneous quadratic constraints we show that a semidefinite programming sdp relaxation for this nonconvex quadratically constrained quadratic program qp provides an o m 2 approximation in the real case and an o m approximation in the complex case moreover we show that these bounds are tight up to a constant factor when the hessian of each constraint function is of rank 1 namely outer products of some given so called steering vectors and the phase spread of the entries of these steering vectors are bounded away from pi 2 we establish a certain constant factor approximation depending on the phase spread but independent of m and n for both the sdp relaxation and a convex qp restriction of the original np hard problem finally we consider a related problem of finding a maximum norm vector subject to m convex homogeneous quadratic constraints we show that an sdp relaxation for this nonconvex qp provides an o 1 ln m approximation which is analogous to a result of nemirovski e math program 86 1999 pp 463 473 for the real case

approximation algorithms for homogeneous polynomial optimization with quadratic constraints in this paper we consider approximation algorithms for optimizing a generic multi variate homogeneous polynomial function subject to homogeneous quadratic constraints such optimization models have wide applications e g in signal processing magnetic resonance imaging mri data training approximation theory and portfolio selection since polynomial functions are non convex the problems under consideration are all np hard in general in this paper we shall focus on polynomial time approximation algorithms in particular we first study optimization of a multi linear tensor function over the cartesian product of spheres we shall propose approximation algorithms for such problem and derive worst case performance ratios which are shown to be dependent only on the dimensions of the model the methods are then extended to optimize a generic multi variate homogeneous polynomial function with spherical constraint likewise approximation algorithms are proposed with provable approximation performance ratios furthermore the constraint set is relaxed to be an intersection of co centered ellipsoids namely we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin and propose polynomial time approximation algorithms with provable worst case performance ratios numerical results are reported illustrating the effectiveness of the approximation algorithms studied