a simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension it has been a challenge for mathematicians to confirm theoretically the extremely good performance of simplex type algorithms for linear programming in this paper the average number of steps performed by a simplex algorithm the so called self dual method is analyzed the algorithm is not started at the traditional point 1 l t but points of the form 1 e e 2 t with e sufficiently small are used the result is better in two respects than those of the previous analyses first it is shown that the expected number of steps is bounded between two quadratic functions c 1 min m n 2 and c 2 min m n 2 of the smaller dimension of the problem this should be compared with the previous two major results in the field borgwardt proves an upper bound of o n 4 m 1 n 1 under a model that implies that the zero vector satisfies all the constraints and also the algorithm under his consideration solves only problems from that particular subclass smale analyzes the self dual algorithm starting at 1 1 t he shows that for any fixed m there is a constant c m such the expected number of steps is less than c m ln n m m 1 megiddo has shown that under smale s model an upper bound c m exists thus for the first time a polynomial upper bound with no restrictions except for nondegeneracy on the problem is proved and for the first time a nontrivial lower bound of precisely the same order of magnitude is established both borgwardt and smale require the input vectors to be drawn from spherically symmetric distributions in the model in this paper invariance is required only under certain

kalman filtering with intermittent observations on the boundedness of the expected error covariance this paper addresses the stability of a kalman filter when measurements are intermittently available due to constraints in the communication channel between the sensor and the estimator we give a necessary condition and a sufficient condition with a trivial gap between them for the boundedness of the expected value of the estimation error covariance these conditions are more general than the existing ones in the sense that they only require the state matrix of the system to be diagonalizable and the sequence of packet losses to be a stationary finite order markov process hence we extend the class of systems for which these conditions are known in two directions namely by including degenerate systems and by considering network models more general than i i d and gilbert elliott we show that these conditions recover known results from the literature when evaluated for non degenerate systems under the assumption of i i d or gilbert elliott packet loss models

birkhoff formula for conformal compressions of symmetric cones in this paper we derive the birkhoff formula for conformal compressions on symmetric cones let v be a euclidean jordan algebra and let be the associated symmetric cone then admits a natural finsler metric contracted by any conformal compressions of we show that the lipschitz constant of a conformal compression of is equal to the hyperbolic tangent of one fourth of the diameter of the image this is the same relation which was obtained by birkhoff on positive reals and by liverani and wojtkowski on the space of positive definite real matrices

weighted maximum likelihood for dynamic factor analysis and forecasting with mixed frequency data for the purpose of forecasting key macroeconomic or financial variables from a panel of time series variables we adopt the dynamic factor model and propose a weighted likelihood based method for parameter estimation the loglikelihood function is split into two parts that are weighted differently the first part is associated with the key variables while the second part is associated with the related variables which may contribute to the forecasting of key variables we derive asymptotic properties including consistency and asymptotic normality of the weighted maximum likelihood estimator we show that this estimator outperforms the standard likelihood based estimator in approximating the true unknown distribution of the data as well as in out of sample forecasting accuracy we verify the new estimation method in a monte carlo study and investigate the role of different weights in different settings in the context of forecasting gross domestic product growth this key variable is typically observed at a low quarterly frequency while the supporting variables are observed at a high monthly frequency we adopt a low frequency representation of the mixed frequency dynamic factor model and discuss the computational efficiencies of this approach in our empirical study for the u s economy we present improvements in nowcasting and forecasting accuracy when the weighted likelihood based estimation procedure is adopted

on the average speed of lemke s algorithm for quadratic programming in this paper we show that the average number of steps of the lemke algorithm for the quadratic programming problems grows at most linearly in the number of variables while fixing the number of constraints the result and method were motivated by smale s result on linear programming problems cf 4 we also give the probability that a quadratic programming problem indeed possesses a finite optimal solution

toward probabilistic analysis of interior point algorithms for linear programming we propose an approach based on interior point algorithms for linear programming lp we show that the algorithm solves a class of lp problems in strongly polynomial time o n log n iteration where each iteration solves a system of linear equations with n variables the recent statistical data of the solutions of the netlib lp problems seem to indicate that virtually all of these problems are in this class then we show that some random lp problems with high probability probability converges to one as n approaches infinity are in this class these random lp problems include recent todd s probabilistic models with the standard gauss distribution

asymptotics for parametric garch in mean models abstract in this paper we develop an asymptotic theory for the quasi maximum likelihood estimator qmle of the parametric garch in mean model the asymptotics is based on a study of the volatility as a process of the model parameters the proof makes use of stochastic recurrence equations for this random function and uses exponential inequalities to localize the problem our results show why the asymptotics for this specification is quite complex although it is a rather standard parametric model nevertheless our theory does not yet treat all standard specifications of the mean function

on the average number of steps of the simplex method of linear programming the goal is to give some theoretical explanation for the efficiency of the simplex method of george dantzig fixing the number of constraints and using dantzig s self dual parametric algorithm we show that the number of pivots required to solve a linear programming problem grows in proportion to the number of variables on the average

fast finite methods for a system of linear inequalities abstract 1 several algorithms are known that solve a system of m linear inequalities in n variables using a polynomial number of steps in m and n for all inputs the simplex method runs in polynomial time in the average case proven and requires only linear in m number of steps in practice we consider a family of finite algorithms some of them involve o m n log3 n iteration steps in the average case under a certain additional assumption about the behavior of the computed approximations to a solution and only o m m n arithmetical operations are required per step the theoretical study is confirmed by some numerical experiments also two methods are presented that enable us to turn any converging algorithm for a system of linear inequalities with real inputs into an algorithm that converges in a finite number of steps

kalman filtering with random coefficients and contractions the riccati transformation of linear filtering control theory is shown to be a contraction on the space of positive symmetric matrices this is used to describe the asymptotic behavior of the filter for systems with stochastic stationary parameters

kalman filtering with intermittent observations convergence for semi markov chains and an intrinsic performance measure this technical note shows that the stationary distribution for the covariance of kalman filtering with intermittent observations exists under mild conditions for a very general class of packet dropping models semi markov chain these results are proved using the geometric properties of riccati recursions with respect to a particular riemannian distance moreover the riemannian mean induced by that distance is always bounded therefore it can be used for characterizing the performance of the system for regimes where the moments of the covariance do not exist other interesting properties of that mean include the symmetry between covariance and information matrices averaging covariances or their inverse gives the same result and its interpretation in information geometry as the natural mean for the manifold of gaussian distributions

new results on the average behavior of simplex algorithms it has been a challenge for mathematicians to theoreti cally confirm the extremely good performance of simplex algorithms for linear programming we have confirmed that a certain variant of the simplex method solves problems of order m x n in an expected num ber of steps which is bounded between two quadratic functions of the smaller dimension of the problem our probabilistic assumptions are rather weak 1 introduction we consider the linear programming problem of order m x n in the form maximize ctx over all x 0 in rn such that ax 5 6 where a e rmxn it has been observed that the simplex algorithms for linear programming developed by george dantzig dl work extremely well a central question in the field of analysis of algorithms is to estimate the expected number of steps that these algorithms perform relative to different probability distributions of inputs some background is given in 52 in am1 and t we consider a variant of the simplex method called the lexicographic self dual method we find that the expected number of steps of this variant relative to a rather weak probabilistic model is only o min m n 2 this is the first polynomial average case upper bound for a simplex algorithm which is capable of solving any linear programming problem the first two authors am21 have also determined a quadratic lower bound so the behavior of this variant is indeed quadratic whereas it has been conjectured that other variants take a linear expected number of steps with various choices of the starting point this variant can simulate different constraint by constraint and variable dimension algorithms as well as combinations thereof me2 adler karp and shamir aks provide a different proof of similar upper bound for a certain class of constraint by constraint algorithms it follows from me21 that their variant can also be simulated by the self dual algorithm with an appropriate starting point

reconstructing polygons from moments with connections to array processing we establish a set of results showing that the vertices of any simply connected planar polygonal region can be reconstructed from a finite number of its complex moments these results find applications in a variety of apparently disparate areas such as computerized tomography and inverse potential theory where in the former it is of interest to estimate the shape of an object from a finite number of its projections whereas in the latter the objective is to extract the shape of a gravitating body from measurements of its exterior logarithmic potentials at a finite number of points we show that the problem of polygonal vertex reconstruction from moments can in fact be posed as an array processing problem and taking advantage of this relationship we derive and illustrate several new algorithms for the reconstruction of the vertices of simply connected polygons from moments

gaussian distributions on riemannian symmetric spaces statistical learning with structured covariance matrices the riemannian geometry of covariance matrices has been essential to several successful applications in computer vision biomedical signal and image processing and radar data processing for these applications an important ongoing challenge is to develop riemannian geometric tools which are adapted to structured covariance matrices this paper proposes to meet this challenge by introducing a new class of probability distributions gaussian distributions of structured covariance matrices these are riemannian analogs of gaussian distributions which only sample from covariance matrices having a preassigned structure such as complex toeplitz or block toeplitz the usefulness of these distributions stems from three features 1 they are completely tractable analytically or numerically when dealing with large covariance matrices 2 they provide a statistical foundation to the concept of structured riemannian barycentre i e frechet or geometric mean and 3 they lead to efficient statistical learning algorithms which realise among others density estimation and classification of structured covariance matrices this paper starts from the observation that several spaces of structured covariance matrices considered from a geometric point of view are riemannian symmetric spaces accordingly it develops an original theory of gaussian distributions on riemannian symmetric spaces of their statistical inference and of their relationship to the concept of riemannian barycentre then it uses this original theory to give a detailed description of gaussian distributions of three kinds of structured covariance matrices complex toeplitz and block toeplitz finally it describes algorithms for density estimation and classification of structured covariance matrices based on gaussian distribution mixture models

frequency response algorithms for h sub infinity optimization with time domain constraints a very broad framework for control system design is considered that encompasses frequency response methodologies for h sub infinity optimization that solve various aspects of the control design problem and that are less well known that state space methods the focus is on linear programming lawson s algorithm and trefethen s algorithm a modified lawson s algorithm is proposed and related to trefethan s method the modified algorithm is shown to be significantly faster than linear programming and lawson s algorithm it is also shown how to extend the modified lawson s algorithm so as to handle time domain constraints in addition to frequency domain specifications which distinguishes it from other h sub infinity optimization methods some steps are taken toward dealing with time domain constraints within an h sub infinity optimization framework

random polytopes their definition generation and aggregate properties the definition of random polytope adopted in this paper restricts consideration to those probability measures satisfying two properties first the measure must induce an absolutely continuous distribution over the positions of the bounding hyperplanes of the random polytope and second it must result in every point in the space being equally as likely as any other point of lying within the random polytope an efficient monte carlo method for their computer generation is presented together with analytical formulas characterizing their aggregate properties in particular it is shown that the expected number of extreme points for such random polytopes increases monotonically in the number of constraints to the limiting case of a polytope topologically equivalent to a hypercube the implied upper bound of 2n wheren is the dimensionality of the space is significantly less than mcmullen s attainable bound on the maximal number of vertices even for a moderate number of constraints

on the solution of the nonlinear matrix equation xn f x abstract we consider a class of nonlinear matrix equations x n f x 0 where f is a self map on the convex cone p k of k k positive definite real matrices it is shown that for n 2 the matrix equation has a unique positive definite solution depending continuously on the function f if f belongs to the semigroup of nonexpansive mappings with respect to the gl k r invariant riemannian metric distance on p k which contains congruence transformations translations the matrix inversion and in particular symplectic hamiltonians appearing in kalman filtering we show that the sequence of positive definite solutions varying over n 2 converges always to the identity matrix

asymptotic behavior of recursive state estimations with intermittent measurements convergence rate and stationary distribution approximation are investigated in this paper for both the covariance matrix of the kalman filter and the pseudo covariance matrix of a robust recursive state estimator developed by us when the measurement dropping process is described by a markov chain and an associated plant is both controllable and observable it is proved that if the dropping probability is less than 1 these two matrices converge exponentially to a stationary distribution independent of its initial values moreover when they are initialized with the stabilizing solution of an associated algebraic riccati equation both of them are shown to converge to an ergodic process based on these results two approximations are derived for their stationary probability distributions as well as a bound of approximation errors it has been made clear that rather than the delta function suggested by kar and moura a series of delta functions give a more accurate approximation and replacement of these estimators update gain by a constant matrix usually deteriorates their steady estimation accuracy a numerical example is provided to illustrate the obtained theoretical results

a family of simplex variants solving an m d linear program in expected number of pivot steps depending on d only we present a family of variants of the simplex method which are based on a constraint by constraint procedure the solution to a linear program is obtained by solving a sequence of subproblems with an increasing number of constraints we discuss several probabilistic models for generating linear programs in all of them the underlying distribution is assumed to be invariant under changing the signs of rows or columns in the problem data a weak regularity condition is also assumed under these models for linear programs with d variables and m d inequality constraints the expected number of pivots required by these algorithms is bounded by a function of minm d only in particular this means that for a fixed number of variables the expected number of pivots is bounded by a constant when the number of constraints tends to infinity since smale s original model smale s 1983 on the average speed of the simplex method of linear programming math programming27 satisfies our probabilistic assumptions the same results apply to his model although not to the particular algorithm he analyzes we also present some results for models generating only feasible linear programs and for bland s pivoting rule we conclude with a discussion of our probabilistic models and show why they are inadequate for obtaining meaningful results unless d and m are of the same order of magnitude

fourier gauss fraunhofer porod and the shape from moments problem we show how the fourier transform of a shape in any number of dimensions can be simplified using gauss s law and evaluated explicitly for polygons in two dimensions polyhedra in three dimensions etc we also show how this combination of fourier and gauss can be related to numerous classical problems in physics and mathematics examples include fraunhofer diffraction patterns porod s law the shape from moments problem and davis s extension of the motzkin schoenberg formula to polygons in the complex plane

diversity effects in kalman filtering over rayleigh fading channels a scalar first order gauss markov signal is sent over d parallel i i d rayleigh fading channels using uncoded transmission assuming perfect channel estimation at the receiver the optimal mmse estimator is the kalman filter with random instantaneous estimation error variance statistical characteristics of the instantaneous estimation error variance are studied to characterize the estimation quality one such characteristic is the estimation error outage which probabilistically depicts how often the value of the instantaneous estimation error variance exceeds a certain threshold in this paper we study the effect of channel diversity on the estimation error outage probability we show that for a certain range of outage thresholds the estimation error outage probability decreases with inverse of d th power of the average signal to noise ratio at the receiver side this means that analogous to the context of digital communication over fading channels our result implies a diversity order of d in estimating the input process

robust recursive state estimation with random measurement droppings a recursive state estimation procedure is derived for a linear time varying system with both parametric uncertainties and stochastic measurement droppings this estimator has a similar form as that of kalman filter with intermittent observations but its parameters should be adjusted when a plant output measurement arrives a new recursive form is derived for the pseudo covariance matrix of estimation errors based on a riemannian metric for positive definite matrices some necessary and sufficient conditions have been obtained for the strict contractiveness of this recursion it has also been proved that under some controllability and observability conditions as well as some weak requirements on measurement arrival probability the update gain of this recursive robust state estimator and the mean of its squared estimation errors converge in probability one respectively to a corresponding stationary distribution numerical simulation results show that estimation accuracy of the suggested procedure is more robust against parametric modelling errors than kalman filter

asymptotic normality of the quasi maximum likelihood estimator for multidimensional causal processes strong consistency and asymptotic normality of the quasi maximum likelihood estimator qmle are given for a general class of multidimensional causal processes for particular cases already studied in the literature for instance univariate or multivariate garch arch arma garch processes the assumptions required for establishing these results are often weaker than existing conditions the qmle asymptotic behavior is also given for numerous new examples of univariate or multivariate processes for instance tarch or nlarch processes

a probabilistic analysis of a measure of combinatorial complexity for the central curve we investigate certain combinatorial properties of the central curve associated with interior point methods for linear optimization we define a measure of complexity for the curve in terms of the number of turns or changes of direction that it makes in a geometric sense and then perform an average case analysis of this measure for p matrix linear complementarity problems we show that the expected number of nondegenerate turns taken by the central curve is bounded by n 2 n where the expectation is taken with respect to a sign invariant probability distribution on the problem data as an alternative measure of complexity we also consider the number of times the central curve intersects with a wide class of algebraic hypersurfaces including such objects as spheres and boxes as an example of the results obtained we show that the primal and dual variables in each coordinate of the central curve cross each other at most once on average as a further example we show that the central curve intersects any sphere centered at the origin at most twice on average

efficiency of the primal network simplex algorithm for the minimum cost circulation problem we study the number of pivots required by the primal network simplex algorithm to solve the minimum cost circulation problem we propose a pivot selection rule with a bound of n logn 2 o 1 on the number of pivots for an n vertex network this is the first known subexponential bound the network simplex algorithm with this rule can be implemented to run in n logn 2 o 1 time in the special case of planar graphs we obtain a polynomial bound on the number of pivots and the running time we also consider the relaxation of the network simplex algorithm in which cost increasing pivots are allowed as well as cost decreasing ones for this algorithm we propose a pivot selection rule with a bound of o nm min log nc m log n on the number of pivots for a network with n vertices m arcs and integer arc costs bounded in magnitude by c the total running time is o nm log n min log nc m log n this bound is within a logarithmic factor of those of the best previously known algorithms for the minimum

convergence of nonlinear observers on bbr n with a riemannian metric part i we study how convergence of an observer whose state lives in a copy of the given system s space can be established using a riemannian metric we show that the existence of an observer guaranteeing the property that a riemannian distance between system and observer solutions is nonincreasing implies that the lie derivative of the riemannian metric along the system vector field is conditionally negative moreover we establish that the existence of this metric is related to the observability of the system s linearization along its solutions moreover if the observer has an infinite gain margin then the level sets of the output function are geodesically convex conversely we establish that if a complete riemannian metric has a lie derivative along the system vector field that is conditionally negative and is such that the output function has a monotonicity property then there exists an observer with an infinite gain margin

on the expected number of linear complementarity cones intersected by random and semi random rays lemke s algorithm for the linear complementarity problem follows a ray which leads from a certain fixed point traditionally the point 1 1 t to the point given in the problem the problem also induces a set of 2 n cones and a question which is relevant to the probabilistic analysis of lemke s algorithm is to estimate the expected number of times a semi random ray intersects the boundary between two adjacent cones when the problem is sampled from a spherically symmetric distribution this number turns out to be exponential for ann dimensional problem the natural logarithm of this number is equal to ln n o n where is approximately 1 151222 this number stands in sharp contrast with the expected number of cones intersected by a ray which is determined by two random points call itrandom the latter is only n 2 1 the discrepancy between linear behavior under the random assumption and exponential behavior under the semi random assumption has implications with respect to recent analyses of the average complexity of the linear programming problem surprisingly the semi random case is very sensitive to the fixed point of the ray even when that point is confined to the positive orthant we show that for points of the form e e 2 e n t the expected number of facets of cones cut by a semi random ray tends to 1 8n 2 3 8n whene tends to zero

quantitative simulations by matrices abstract we introduce notions of simulation between semiring weighted automata as models of quantitative systems our simulations are instances of the categorical coalgebraic notions previously studied by hasuo hence soundness against language inclusion comes for free but are concretely presented as matrices that are subject to linear inequality constraints pervasiveness of these formalisms allows us to exploit existing algorithms in searching for a simulation and hence verifying quantitative correctness that is formulated as language inclusion transformations of automata that aid search for simulations are introduced too this verification workflow is implemented for the plus times and max plus semirings furthermore an extension to weighted tree automata is presented and implemented

the efficiency of the simplex method a survey the linear programming problem is by far the most widely used optimization model its impact on economic and government modeling is immense the simplex method for solving the linear programming lp problem due to george dantzig has been an extremely efficient computational tool for almost four decades the method has been the subject of intense investigations for many years but some major aspects of its behavior are not fully understood yet the purpose of this paper is to survey the body of knowledge on the efficiency of the simplex method from both practical and theoretical points of view adopting the number of iterations pivot steps as the yardstick for efficiency we survey four aspects of the issue 1 reports on practical experience of the method s performance on real life lp problems 2 results on controlled monte carlo experiments solving lp problems which were randomly generated according to some predetermined distributions 3 complexity results including theoretical analyses on both upper and lower bounds for the performance of the simplex as well as non simplex algorithms for lp 4 results of recent theoretical studies using probabilistic analysis to derive bounds on the average behavior of the simplex method we discuss the consequences and limitations of the various studies special emphasis is given to open questions

average polynomial time complexity of some np complete problems abstract we define clique n n to be the problem clique restricted to the set of graphs g v e such that e n v in this paper it is proved that when n n n e e a rational number 0 clique n n is an np complete problem and under the assumption of a uniform distribution on the set of instances clique n n can be solved by a deterministic algorithm in average and almost everywhere polynomial time some cases of nonuniform distributions on the set of instances for which the usual problem clique is solved in average polynomial time are also considered

mean square stability for kalman filtering with markovian packet losses this paper studies the stability of kalman filtering over a network subject to random packet losses which are modeled by a time homogeneous ergodic markov process for second order systems necessary and sufficient conditions for stability of the mean estimation error covariance matrices are derived by taking into account the system structure while for certain classes of higher order systems necessary and sufficient conditions are also provided to ensure stability of the mean estimation error covariance matrices all stability criteria are expressed by simple inequalities in terms of the largest eigenvalue of the open loop matrix and transition probabilities of the markov process their implications and relationships with related results in the literature are discussed

nonexpansiveness of the resolvent average abstract we show that the resolvent average for positive definite matrices which interpolates between the harmonic and the arithmetic means contracts the thompson metric on the convex cone of positive definite matrices classical results depending on the nonexpansive property of the arithmetic average are considered in the context of the resolvent average in particular resolvent power mean approximation to the karcher mean and ferrante and levy like matrix equations are investigated

mutual information of wireless channels and block jacobi ergodic operators shannon s mutual information of a random multiple antenna and multipath time varying channel is studied in the general case where the process constructed from the channel coefficients is an ergodic and stationary process which is assumed to be available at the receiver from this viewpoint the channel can also be represented by an ergodic self adjoint block jacobi operator which is close in many aspects to a block version of a random schrodinger operator the mutual information is then related to the so called density of states of this operator in this paper it is shown that under the weakest assumptions on the channel the mutual information can be expressed in terms of a matrix valued stochastic process coupled with the channel process this allows numerical approximations of the mutual information in this general setting moreover assuming further that the channel coefficient process is a markov process a representation for the mutual information offset in the large signal to noise ratio regime is obtained in terms of another related markov process this generalizes previous results from levy et al it is also illustrated how the mutual information expressions that are closely related to those predicted by the random matrix theory can be recovered in the large dimensional regime

exact semidefinite formulations for a class of random and non random nonconvex quadratic programs we study a class of quadratically constrained quadratic programs qcqps called diagonal qcqps which contain no off diagonal terms x j x k for j ne k and we provide a sufficient condition on the problem data guaranteeing that the basic shor semidefinite relaxation is exact our condition complements and refines those already present in the literature and can be checked in polynomial time we then extend our analysis from diagonal qcqps to general qcqps i e ones with no particular structure by reformulating a general qcqp into diagonal form we establish new polynomial time checkable sufficient conditions for the semidefinite relaxations of general qcqps to be exact finally these ideas are extended to show that a class of random general qcqps has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables to the best of our knowledge this is the first result establishing the exactness of the semidefinite relaxation for random general qcqps

a stable numerical method for inverting shape from moments we derive a stable technique based upon matrix pencils for the reconstruction of or approximation by polygonal shapes from moments we point out that this problem can be considered the dual of 2 d numerical quadrature over polygonal domains an analysis of the sensitivity of the problem is presented along with some numerical examples illustrating the relevant points finally an application to the problem of gravimetry is explored where the shape of a gravitationally anomalous region is to be recovered from measurements of its exterior gravitational field

hard to solve bimatrix games the lemke howson algorithm is the classical method for finding one nash equilibrium of a bimatrix game this paper presents a class of square bimatrix games for which this algorithm takes even in the best case an exponential number of steps in the dimension d of the game using polytope theory the games are constructed using pairs of dual cyclic polytopes with 2d suitably labeled facets in d space the construction is extended to nonsquare games where in addition to exponentially long lemke howson computations finding an equilibrium by support enumeration takes on average exponential time

improved asymptotic analysis of the average number of steps performed by the self dual simplex algorithm in this paper we analyze the average number of steps performed by the self dual simplex algorithm for linear programming under the probabilistic model of spherical symmetry the model was proposed by smale consider a problem ofn variables withm constraints smale established that for every number of constraintsm there is a constantc m such that the number of pivot steps of the self dual algorithm m n is less thanc m lnn m m 1 we improve upon this estimate by showing that m n is bounded by a function ofm only the symmetry of the function inm andn implies that m n is in fact bounded by a function of the smaller ofm andn

random linear programs with many variables and few constraints we extend and simplify smale s work on the expected number of pivots for a linear program with many variables and few constraints our analysis applies to new versions of the simplex algorithm and to new random distributions

multidimensional integral inversion with applications in shape reconstruction in shape reconstruction the celebrated fourier slice theorem plays an essential role it allows one to reconstruct the shape of a quite general object from the knowledge of its radon transform s helgason the radon transform birkhauser boston boston 1980 in other words from the knowledge of projections of the object in case the object is a polygon g h golub p milanfar and j varah siam j sci comput 21 1999 pp 1222 1243 or when it defines a quadrature domain in the complex plane b gustafsson c he p milanfar and m putinar inverse problems 16 2000 pp 1053 1070 its shape can also be reconstructed from the knowledge of its moments essential tools in the solution of the latter inverse problem are quadrature rules and formal orthogonal polynomials in this paper we show how shape reconstruction from the knowledge of moments can also be realized in the case of general compact objects not only in two but also in higher dimensions to this end we use a less known homogeneous pade slice property again integral transforms in our case the multivariate stieltjes transform and univariate markov transform formal orthogonal polynomials in the form of pade denominators and multidimensional integration formulas or cubature rules play an essential role we emphasize that the new technique is applicable in all higher dimensions and illustrate it through the reconstruction of several two and three dimensional objects

stochastic stability of the discrete time extended kalman filter the authors analyze the error behavior for the discrete time extended kalman filter for general nonlinear systems in a stochastic framework in particular it is shown that the estimation error remains bounded if the system satisfies the nonlinear observability rank condition and the initial estimation error as well as the disturbing noise terms are small enough this result is verified by numerical simulations for an example system

performance of ensemble kalman filters in large dimensions contemporary data assimilation often involves more than a million prediction variables ensemble kalman filters enkf have been developed by geoscientists they are successful indispensable tools in science and engineering because they allow for computationally cheap low ensemble state approximation for extremely large dimensional turbulent dynamical systems the practical finite ensemble filters like enkf necessarily involve modifications such as covariance inflation and localization and it is a genuine mystery why they perform so well with small ensemble sizes in large dimensions this paper provides the first rigorous stochastic analysis of the accuracy and covariance fidelity of enkf in the practical regime where the ensemble size is much smaller than the large ambient dimension for enkfs with random coefficients a challenging issue overcome here is that enkf in huge dimensions introduces unavoidable bias and model errors that need to be controlled and estimated 2017 the authors communications on pure and applied mathematics is published by the courant institute of mathematics and wiley periodicals inc

the d step conjecture and its relatives the d step conjecture arose from an attempt to understand the computational complexity of edge following algorithms for linear programming such as the simplex algorithm it can be stated in terms of diameters of graphs of convex polytopes in terms of the existence of nonrevisiting paths in such graphs in terms of an exchange process for simplicial bases of a vector space and in terms of matrix pivot operations first formulated by w m hirsch in 1957 the conjecture remains unsettled though it has been proved in many special cases and counterexamples have been found for slightly stronger conjectures if the conjecture is false as we believe to be the case then finding a counterexample will be merely a small first step in the line of investigation related to the conjecture this report summarizes what is known about the d step conjecture and its relatives a considerable amount of new material is included but it does not seem to come close to settling the conjecture of special interest is the first example of a polytope that is not vertex decomposable showing that a certain natural approach to the conjecture will not work also significant are the quantitative relations among the lengths of paths associated with various forms of the conjecture

a counterexample to the hirsch conjecture the hirsch conjecture 1957 stated that the graph of a d dimensional polytope with n facets cannot have combinatorial diameter greater than n d that is any two vertices of the polytope can be connected by a path of at most n d edges this paper presents the rst counterexample to the conjecture our polytope has dimension 43 and 86 facets it is obtained from a 5 dimensional polytope with 48 facets that violates a certain generalization of the d step conjecture of klee and walkup

gossip and distributed kalman filtering weak consensus under weak detectability the paper presents the gossip interactive kalman filter gikf for distributed kalman filtering for networked systems and sensor networks where intersensor communication and observations occur at the same time scale the communication among sensors is random each sensor occasionally exchanges its filtering state information with a neighbor depending on the availability of the appropriate network link we show that under a weak distributed detectability condition 1 the gikf error process remains stochastically bounded irrespective of the instability of the random process dynamics and 2 the network achieves weak consensus i e the conditional estimation error covariance at a uniformly randomly selected sensor converges in distribution to a unique invariant measure on the space of positive semidefinite matrices independent of the initial state to prove these results we interpret the filtering states estimates and error covariances at each node in the gikf as stochastic particles with local interactions we analyze the asymptotic properties of the error process by studying as a random dynamical system the associated switched random riccati equation the switching being dictated by a nonstationary markov chain on the network graph

the contraction rate in thompson s part metric of order preserving flows on a cone application to generalized riccati equations abstract we give a formula for the lipschitz constant in thompson s part metric of any order preserving flow on the interior of a possibly infinite dimensional closed convex pointed cone this shows that in the special case of order preserving flows a general characterization of the contraction rate in thompson s part metric given by nussbaum leads to an explicit formula as an application we show that the flow of the generalized riccati equation arising in stochastic linear quadratic control is a local contraction on the cone of positive definite matrices and characterize its lipschitz constant by a matrix inequality we also show that the same flow is no longer a contraction in other invariant finsler metrics on this cone including the standard invariant riemannian metric this is motivated by a series of contraction properties concerning the standard riccati equation established by bougerol liverani wojtkowski lawson lee and lim we show that some of these properties do and that some other do not carry over to the generalized riccati equation

a sharp upper bound for the expected number of shadow vertices in lp polyhedra under orthogonal projection on two dimensional planes consider a set a1 am u v n satisfying a mild nondegeneracy condition when the polyhedron x x a1tx 1 amtx 1 n is orthogonally projected on the two dimensional plane linu v then some of its vertices may be mapped on vertices of the two dimensional image of x these vertices of x will be called shadow vertices under that projection their number will be denoted by s this figure s has an outstanding relevance for the complexity of a special simplex method the dimension by dimension method for solving lps of the type maximize vtx s t a1tx 1 amtx 1 where x v a1 am n and m n this method applies the shadow vertex algorithm in n 1 stages of increasing dimension by constructing a simplex path that visits only shadow vertices in each stage specific versions of the vectors u and v are used to define the projection plane and to control the application of the shadow vertex algorithm so s delivers an upper bound for the length of such a path in one stage we want to study the average case complexity of these algorithms and are therefore interested in em ns the expected value of s for that purpose we assume a distribution of the linear programming problems corresponding to the rotation symmetry model rsm the vectors a1 am u v are distributed on n 0 independently identically and symmetrically under rotations in this paper we improve the upper bounds for em ns from our polynomiality proofs of 1982 and 1987 there we had found an upper bound of m1 n 1 i n3 i const valid for all rsm distributions and all pairs m n based on a refined evaluation technique for space angles on a reorganization and on substantial modifications of the old proof we are now able to confirm a bound em ns m1 n 1 i n2 i const for all pairs m n with m n this result had been desired since 1982 because then a bound m1 n 1 i n2 i const had been derived exclusively for the situation where n is fixed and m tends to infinity all the time the question had been open whether the asymptotical behaviour would be better than the behaviour in moderate dimensions or whether the bound for the moderate case was crude in addition we know from an asymptotic analysis in 1987 that for a special rsm distribution namely the uniform distribution on the unit sphere there is a lower bound for em ns of type m1 n 1 i n2 i const the combination of this result and the new upper bound shows that the new bound is sharp

the many facets of linear programming we examine the history of linear programming from computational geometric and complexity points of view looking at simplex ellipsoid interior point and other methods

polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming problems we show that a particular pivoting algorithm which we call the lexicographic lemke algorithm takes an expected number of steps that is bounded by a quadratic inn when applied to a random linear complementarity problem of dimensionn we present two probabilistic models both requiring some nondegeneracy and sign invariance properties the second distribution is concerned with linear complementarity problems that arise from linear programming in this case we give bounds that are quadratic in the smaller of the two dimensions of the linear programming problem and independent of the larger similar results have been obtained by adler and megiddo

a quadratically convergent method for linear programming abstract a new method to solve linear programming problems is introduced this method follows a path defined by a system of o d e and for nondegenerate problem is quadratically convergent

brief paper optimal estimation of linear discrete time systems with stochastic parameters this paper considers optimal linear state estimation in the general case of linear discrete time systems with stochastic parameters which are statistically independent with respect to time the estimator is derived by transforming the system to one with deterministic parameters and state dependent additive system and observation noise it is shown that mean square stability of the system is a sufficient and almost necessary condition for the existence uniqueness and stability of the time invariant estimator

the expected number of nash equilibria of a normal form game fix finite pure strategy sets s1 s and let s s x x s n in our model of a random game the agents payoffs are statistically independent with each agent s payoff uniformly distributed on the unit sphere in r a for given nonempty t 1 subset of s 1 t n subset of s n we give a computationally implementable formula for the mean number of nash equilibria in which each agent i s mixed strategy has support t i the formula is the product of two expressions the first is the expected number of totally mixed equilibria for the truncated game obtained by eliminating pure strategies outside the sets ti the second may be construed as the probability that such an equilibrium remains an equilibrium when the strategies in the sets s i t i become available

random inequality constraint systems with few variables we show that a system of many linear inequality constraints will have a high proportion of redundant constraints with high probability implications for expected time of algorithms are indicated

probabilistic models for linear programming we propose and investigate new probabilistic models for linear programming in contrast to previous models ours guarantee the existence of optimal solutions and are symmetric under duality while in some respects our distributions are very special there is sufficient flexibility to permit an arbitrary degree of primal and or dual degeneracy either just at the optimal solution or throughout the feasible region using null variables moreover the precision of the distributions allows us to compute the probability that the feasible region is bounded as well as the distribution of the distance to a constraint hyperplane and that of the components of a vertex interest in these measures stems from karmarkar s algorithm and we also introduce a model for generating random linear programming problems on a simplex

parametric linear programming and anti cycling pivoting rules the traditional perturbation or lexicographic methods for resolving degeneracy in linear programming impose decision rules that eliminate ties in the simplex ratio rule and therefore restrict the choice of exiting basic variables bland s combinatorial pivoting rule also restricts the choice of exiting variables using ideas from parametric linear programming we develop anticycling pivoting rules that do not limit the choice of exiting variables beyond the simplex ratio rule that is any variable that ties for the ratio rule can leave the basis a similar approach gives pivoting rules for the dual simplex method that do not restrict the choice of entering variables