convergence of time stepping schemes for passive and extended linear complementarity systems generalizing recent results in m k camlibel complementarity methods in the analysis of piecewise linear dynamical systems ph d thesis center for economic research tilburg university tilburg the netherlands 2001 m k camlibel w p m h heemels and j m schumacher ieee trans circuits systems i fund theory appl 49 2002 pp 349 357 and j s pang and d stewart math program ser a 113 2008 pp 345 424 this paper provides an in depth analysis of time stepping methods for solving initial value and boundary value non lipschitz linear complementarity systems lcss under passivity and broader assumptions the novelty of the methods and their analysis lies in the use of least norm solutions in the discrete time linear complementarity subproblems arising from the numerical scheme these subproblems are not necessarily monotone and are not guaranteed to have convex solution sets among the principal results it is shown that using such least norm solutions of the discrete time subproblems an implicit euler scheme is convergent for passive initial value lcss generalizations under a strict copositivity assumption and for boundary value lcss are also established

strongly regular differential variational systems a differential variational system is defined by an ordinary differential equation ode parameterized by an algebraic variable that is required to be a solution of a finite dimensional variational inequality containing the state variable of the system this paper addresses two system theoretic topics for such a nontraditional nonsmooth dynamical system namely non zenoness and local observability of a given state satisfying a blanket strong regularity condition for the former topic which is of contemporary interest in the study of hybrid systems we extend the results in our previous paper where we have studied zeno states and switching times in a linear complementarity system lcs as a special case of the differential variational inequality dvi the lcs consists of a linear time invariant ode and a linear complementarity problem the extension to a nonlinear complementarity system ncs with analytic inputs turns out to be non trivial as we need to use the lie derivatives of analytic functions in order to arrive at an expansion of the solution trajectory near a given state further extension to a differential variational inequality is obtained via its equivalent karush kuhn tucker formulation for the second topic which is classical in system theory we use the non zenoness result and the recent results in a previous paper pertaining to the b differentiability of the solution operator of a nonsmooth ode to obtain a sufficient condition for the short time local observability of a given strongly regular state of an ncs refined sufficient conditions and necessary conditions for local observability of the lcs satisfying the p property are obtained

differential variational inequalities this paper introduces and studies the class of differential variational inequalities dvis in a finite dimensional euclidean space the dvi provides a powerful modeling paradigm for many applied problems in which dynamics inequalities and discontinuities are present examples of such problems include constrained time dependent physical systems with unilateral constraints differential nash games and hybrid engineering systems with variable structures the dvi unifies several mathematical problem classes that include ordinary differential equations odes with smooth and discontinuous right hand sides differential algebraic equations daes dynamic complementarity systems and evolutionary variational inequalities conditions are presented under which the dvi can be converted either locally or globally to an equivalent ode with a lipschitz continuous right hand function for dvis that cannot be so converted we consider their numerical resolution via an euler time stepping procedure which involves the solution of a sequence of finite dimensional variational inequalities borrowing results from differential inclusions dis with upper semicontinuous closed and convex valued multifunctions we establish the convergence of such a procedure for solving initial value dvis we also present a class of dvis for which the theory of dis is not directly applicable and yet similar convergence can be established finally we extend the method to a boundary value dvi and provide conditions for the convergence of the method the results in this paper pertain exclusively to systems with index not exceeding two and which have absolutely continuous solutions

a gauss seidel type method for dynamic nonlinear complementarity problems the dynamic nonlinear complementarity problem dncp consisting of a nonlinear differential system and a complementarity system has been used to formulate and study many dynamic problems in a gaus

an lpcc approach to nonconvex quadratic programs filling a gap in nonconvex quadratic programming this paper shows that the global resolution of a feasible quadratic program qp which is not known a priori to be bounded or unbounded below can be accomplished in finite time by solving two linear programs with linear complementarity constraints i e lpccs specifically this task can be divided into two lpccs the first confirms whether the qp is bounded below on the feasible set and if not computes a feasible ray on which the qp is unbounded the second lpcc computes a globally optimal solution if it exists by identifying a stationary point that yields the best quadratic objective value in turn the global resolution of these lpccs can be accomplished by a parameter free mixed integer programming based finitely terminating algorithm developed recently by the authors which can be enhanced in this context by a new kind of valid cut derived from the second order conditions of the qp and by exploiting the special structure of the lpccs throughout our treatment makes no boundedness assumption of the qp this is a significant departure from much of the existing literature which consistently employs the boundedness of the feasible set as a blanket assumption the general theory is illustrated by 3 classes of indefinite problems qps with simple upper and lower bounds existence of optimal solutions is guaranteed same qps with an additional inequality constraint extending the case of simple bound constraints and nonnegatively constrained copositive qps no guarantee of the existence of an optimal solution we also present numerical results to support the special cuts obtained due to the qp connection

mixture densities maximum likelihood and the em algorithm the problem of estimating the parameters which determine a mixture density has been the subject of a large diverse body of literature spanning nearly ninety years during the last two decades the

evaluation of the undetected error probability of single parity check product codes in this paper an expression for the undetected error probability p epsilon of single parity check product spcp codes used for error detection over a binary symmetric channel is derived it is shown that square spcp codes need not obey a certain commonly used bound approximate expressions for the maximum p epsilon and the corresponding maximizing e are given

brief paper a full characterization of stabilizability of bimodal piecewise linear systems with scalar inputs this paper studies open loop stabilization problem for bimodal systems with continuous vector field it is based on the earlier work of the authors on the controllability problem for the same class of systems a full characterization of stabilizability is established by presenting algebraic necessary and sufficient conditions it is also shown that controllability implies stabilizability for these systems in a very similar fashion to the linear case

edge coloring and stopping sets analysis in product codes with mds components we consider non binary product codes with mds components and their iterative row column algebraic decoding on the erasure channel both independent and block erasures are considered in this paper a compact graph representation is introduced on which we define double diversity edge colorings via the rootcheck concept an upper bound of the number of decoding iterations is given as a function of the graph size and the color palette size m then we propose a differential evolution edge coloring algorithm that produces colorings with a large population of minimal rootcheck order symbols the complexity of this algorithm per iteration is o m aleph for a given differential evolution parameter aleph where m aleph itself is small with respect to the huge cardinality of the coloring ensemble stopping sets of a product code are defined in the context of mds components and a relationship is established with the graph representation a full characterization of these stopping sets is given up to a size d 1 2 where d is the minimum hamming distance of the mds component code the performance of mds based product codes with and without double diversity coloring is analyzed in presence of both the block and the independent erasures in the latter case ml and iterative decoding are proven to coincide at small channel erasure probability furthermore numerical results show excellent performance in presence of unequal erasure probability due to double diversity colorings

convergence of regularized time stepping methods for differential variational inequalities this paper provides convergence analysis of regularized time stepping methods for the differential variational inequality dvi which consists of a system of ordinary differential equations and a parametric variational inequality pvi as the constraint the pvi often has multiple solutions at each step of a time stepping method and it is hard to choose an appropriate solution for guaranteeing the convergence in l han a tiwari m k camlibel and j s pang siam j numer anal 47 2009 pp 3768 3796 the authors proposed to use least norm solutions of parametric linear complementarity problems at each step of the time stepping method for the monotone linear complementarity system and showed the novelty and advantages of the use of the least norm solutions however in numerical implementation when the pvi is not monotone and its solution set is not convex finding a least norm solution is difficult this paper extends the tikhonov regularization approximation to the p 0 function dvi whi

uniqueness for solutions of differential complementarity problems in this paper we consider the question of which matrices m give unique solutions for differential complementarity problems mandelbaum 1989 unpublished manuscript of the form begin array ll frac dw dt m z q t quad w 0 w 0 k ni z t perp w t in k quad rm for all t end array with applications to complementarity and discretized l1 regularization problems assuming semi smoothness it is shown that super linearly convergent newton methods can be globalized if appropriate descent directions are used for the merit function f x 2 special attention is paid to directions obtained from the primal dual active set strategy

algebraic necessary and sufficient conditions for the controllability of conewise linear systems the problem of checking certain controllability properties of even very simple piecewise linear systems is known to be undecidable this paper focuses on conewise linear systems i e systems for which the state space is partitioned into conical regions and a linear dynamics is active on each of these regions for this class of systems we present algebraic necessary and sufficient conditions for controllability we also show that the classical results of controllability of linear systems and input constrained linear systems can be recovered from our main result our treatment employs tools both from geometric control theory and mathematical programming

distributed estimation and detection for sensor networks using hidden markov random field models we develop a hidden markov random field hmrf framework for distributed signal processing in sensor network environments under this framework spatially distributed observations collected at the sensors form a noisy realization of an underlying random field that has a simple structure with markovian dependence we derive iterated conditional modes icm algorithms for distributed estimation of the hidden random field from the noisy measurements we consider both parametric and nonparametric measurement error models the proposed distributed estimators are computationally simple applicable to a wide range of sensing environments and localized implying that the nodes communicate only with their neighbors to obtain the desired results we also develop a calibration method for estimating markov random field model parameters from training data and discuss initialization of the icm algorithms the hmrf framework and icm algorithms are applied to event region detection numerical simulations demonstrate the performance of the proposed approach

evolutionary problems driven by variational inequalities abstract in this paper we introduce the differential system obtained by mixing an evolution equation and a variational inequality eevi for short first by using kkm theorem and monotonicity arguments we prove the superpositional measurability and upper semicontinuity for the solution set of a general variational inequality then we establish that the solution set of eevi is nonempty and compact our approach is based on the theory of semigroups filippov implicit function lemma and fixed point theory for set valued mappings

non zenoness of a class of differential quasi variational inequalities the zeno phenomenon of a switched dynamical system refers to the infinite number of mode switches in finite time the absence of this phenomenon is crucial to the numerical simulation of such a system by time stepping methods and to the understanding of the behavior of the system trajectory extending a previous result for a strongly regular differential variational inequality this paper establishes that a certain class of non strongly regular differential variational inequalities is devoid of the zeno phenomenon the proof involves many supplemental results that are of independent interest specialized to a frictional contact problem with local compliance and polygonal friction laws this non zenoness result is of fundamental significance and the first of its kind

coordination and consensus of networked agents with noisy measurements stochastic algorithms and asymptotic behavior this paper considers the coordination and consensus of networked agents where each agent has noisy measurements of its neighbors states for consensus seeking we propose stochastic approximation type algorithms with a decreasing step size and introduce the notions of mean square and strong consensus although the decreasing step size reduces the detrimental effect of the noise it also reduces the ability of the algorithm to drive the individual states towards each other the key technique is to ensure a trade off for the decreasing rate of the step size by following this strategy we first develop a stochastic double array analysis in a two agent model which leads to both mean square and strong consensus and extend the analysis to a class of well studied symmetric models subsequently we consider a general network topology and introduce stochastic lyapunov functions together with the so called direction of invariance to establish mean square consensus finally we apply the stochastic lyapunov analysis to a leader following scenario

a distributed classification estimation algorithm for sensor networks in this paper we address the problem of simultaneous classification and estimation of hidden parameters in a sensor network with communications constraints in particular we consider a network of noisy sensors which measure a common scalar unknown parameter we assume that a fraction of the nodes represent faulty sensors whose measurements are poorly reliable the goal for each node is to simultaneously identify its class faulty or nonfaulty and estimate the common parameter we propose a novel cooperative iterative algorithm which copes with the communication constraints imposed by the network and shows remarkable performance our main result is a rigorous proof of the convergence of the algorithm under a fixed communication graph and a characterization of the limit behavior as the network size goes to infinity in particular we prove that in the limit when the number of sensors goes to infinity the common unknown parameter is estimated with arbitrary small error while the classification error

the joint map ml criterion and its relation to ml and to extended least squares the joint maximum a posteriori maximum likelihood jmap ml estimation criterion can serve as an alternative to the maximum likelihood ml criterion when estimating parameters from an observed data vector whenever another unobserved data vector is involved rather than maximize the probability of the observed data with respect to the parameters jmap ml maximizes the joint probability of the observed and unobserved data with respect to both the unknown parameters and the unobserved data in this paper we characterize the relation between the ml and jmap ml estimates in the gaussian case and provide insight into the apparent bias of jmap ml although jmap ml is an inconsistent estimator we show that with short data records it is often preferable to ml in terms of both bias and variance we also identify jmap ml as a special case of the deterministic extended least squares xls criterion we indicate a general relation between a possible maximization algorithm for jmap ml and the well known estimation maximization em algorithm

differential variational inequality approach to dynamic games with shared constraints the dynamic nash equilibrium problem with shared constraints nepsc involves a dynamic decision process with multiple players where not only the players cost functionals but also their admissible control sets depend on the rivals decision variables through shared constraints for a class of the dynamic nepsc we propose a differential variational inequality formulation using this formulation we show the existence of solutions of the dynamic nepsc and develop a regularized smoothing method to find a solution of it we prove that the regularized smoothing method converges to the least norm solution of the differential variational inequality which is a solution of the dynamic nepsc as the regularization parameter lambda and smoothing parameter mu go to zero with the order mu o lambda o numerical examples are given to illustrate the existence and convergence results

popov belevitch hautus type tests for the controllability of linear complementarity systems it is well known that checking certain controllability properties of very simple piecewise linear systems are undecidable problems this paper deals with the controllability problem of a class of piecewise linear systems known as linear complementarity systems by exploiting the underlying structure and employing the results on the controllability of the so called conewise linear systems we present a set of inequality type conditions as necessary and sufficient conditions for controllability of linear complementarity systems the presented conditions are of popov belevitch hautus type in nature

stopping set analysis of iterative row column decoding of product codes in this paper we introduce stopping sets for iterative row column decoding of product codes using optimal constituent decoders when transmitting over the binary erasure channel bec iterative row column decoding of product codes using optimal constituent decoders will either be successful or stop in the unique maximum size stopping set that is contained in the initial set of erased positions let cp denote the product code of two binary linear codes cc and cr of minimum distances dc and dr and second generalized hamming weights d2 cc and d2 cr respectively we show that the size smin of the smallest noncode word stopping set is at least mm drd2 cc dcd2 cr drdc where the inequality follows from the griesmer bound if there are no codewords in cp with support set s where s is a stopping set then s is said to be a noncodeword stopping set an immediate consequence is that the erasure probability after iterative row column decoding using optimal constituent decoders of finite length product codes on the bec approaches the erasure probability after maximum likelihood decoding as the channel erasure probability decreases we also give an explicit formula for the number of noncodeword stopping sets of size smin which depends only on the first nonzero coefficient of the constituent row and column first and second support weight enumerators for the case when d2 cr 2dr and d2 cc 2dc finally as an example we apply the derived results to the product of two extended hamming codes and two golay codes

the rational complementarity problem abstract an extension of the linear complementarity problem lcp of mathematical programming is the so called rational complementarity problem rcp this problem occurs if complementarity conditions are imposed on input and output variables of linear dynamical input state output systems the resulting dynamical systems are called linear complementarity systems since the rcp is crucial both in issues concerning existence and uniqueness of solutions to complementarity systems and in time simulation of complementarity systems it is worthwhile to consider existence and uniqueness questions of solutions to the rcp in this paper necessary and sufficient conditions are presented guaranteeing existence and uniqueness of solutions to the rcp in terms of corresponding lcps using these results and proving that the corresponding lcps have certain properties we can show uniqueness and existence of solutions to linear mechanical systems with unilateral constraints electrical networks with diodes and linear dynamical systems subject to relays and or coulomb friction

lyapunov stability of complementarity and extended systems a linear complementarity system lcs is a piecewise linear dynamical system consisting of a linear time invariant ordinary differential equation ode parameterized by an algebraic variable that is required to be a solution to a finite dimensional linear complementarity problem lcp whose constant vector is a linear function of the differential variable continuing the authors recent investigation of the lcs from the combined point of view of system theory and mathematical programming this paper addresses the important system theoretic properties of exponential and asymptotic stability for an lcs with a c 1 state trajectory the novelty of our approach lies in our employment of a quadratic lyapunov function that involves the auxiliary algebraic variable of the lcs when expressed in the state variable alone the lyapunov function is piecewise quadratic and thus nonsmooth the nonsmoothness feature invalidates standard stability analysis that is based on smooth lyapunov functions in addition to providing sufficient conditions for exponential stability we establish a generalization of the well known lasalle invariance theorem for the asymptotic stability of a smooth dynamical system to the lcs which is intrinsically a nonsmooth system sufficient matrix theoretic copositivity conditions are introduced to facilitate the verification of the stability properties properly specialized the latter conditions are satisfied by a passive like lcs and certain hybrid linear systems having common quadratic lyapunov functions we provide numerical examples to illustrate the stability results we also develop an extended local exponential stability theory for nonlinear complementarity systems and differential variational inequalities based on a new converse theorem for odes with b differentiable right hand sides the latter theorem asserts that the existence of a b differentiable lyapunov function is a necessary and sufficient condition for the exponential stability of an equilibrium of such a differential system

complementarity modeling of hybrid systems a complementarity framework is described for the modeling of certain classes of mixed continuous discrete dynamical systems the use of such a framework is well known for mechanical systems with inequality constraints but we give a more general formulation which also applies for instance to switching control systems the main theoretical results in the paper are concerned with uniqueness of smooth continuations the solution of this problem requires the construction of a map from the continuous state to the discrete state a crucial technical tool is the so called linear complementarity problem from mathematical programming and we introduce various generalizations of this problem

linear complementarity systems zeno states a linear complementarity system lcs is a hybrid dynamical system defined by a linear time invariant ordinary differential equation coupled with a finite dimensional linear complementarity problem lcp the present paper is the first of several papers whose goal is to study some fundamental issues associated with an lcs specifically this paper addresses the issue of zeno states and the related issue of finite number of mode switches in such a system the cornerstone of our study is an expansion of a solution trajectory to the lcs near a given state in terms of an observability degree of the state on the basis of this expansion and an inductive argument we establish that an lcs satisfying the p property has no strongly zeno states we next extend the analysis for such an lcs to a broader class of problems and provide sufficient conditions for a given state to be weakly non zeno while related mode switch results have been proved by brunovsky and sussmann for more general hybrid systems our analysis exploits the special structure of the lcs and yields new results for the latter that are of independent interest and complement those by these two and other authors

linear complementarity systems we introduce a new class of dynamical systems called linear complementarity systems the time evolution of these systems consists of a series of continuous phases separated by events which cause a change in dynamics and possibly a jump in the state vector the occurrence of events is governed by certain inequalities similar to those appearing in the linear complementarity problem of mathematical programming the framework we describe is suitable for certain situations in which both differential equations and inequalities play a role for instance in mechanics electrical networks piecewise linear systems and dynamic optimization we present a precise definition of the solution concept of linear complementarity systems and give sufficient conditions for existence and uniqueness of solutions

more results on the weight enumerator of product codes we consider the product code c sub p of q ary linear codes with minimum distances d sub c and d sub r the words in c sub p of weight less than d sub r d sub c max d sub r spl lceil d sub c g spl rceil d sub c spl lceil d sub r q spl rceil are characterized and their number is expressed in the number of low weight words of the constituent codes for binary product codes we give an upper bound on the number of words in c sub p of weightless than min d sub r d sub c spl lceil d sub c 2 spl rceil 1 d sub c d sub r spl lceil d sub r 2 spl rceil 1 that is met with equality if c sub c and c sub r are extended perfect codes

distributed em algorithm for gaussian mixtures in sensor networks this paper presents a distributed expectation maximization em algorithm over sensor networks in the e step of this algorithm each sensor node independently calculates local sufficient statistics by using local observations a consensus filter is used to diffuse local sufficient statistics to neighbors and estimate global sufficient statistics in each node by using this consensus filter each node can gradually diffuse its local information over the entire network and asymptotically the estimate of global sufficient statistics is obtained in the m step of this algorithm each sensor node uses the estimated global sufficient statistics to update model parameters of the gaussian mixtures which can maximize the log likelihood in the same way as in the standard em algorithm because the consensus filter only requires that each node communicate with its neighbors the distributed em algorithm is scalable and robust it is also shown that the distributed em algorithm is a stochastic approximation to the standard em algorithm thus it converges to a local maximum of the log likelihood several simulations of sensor networks are given to verify the proposed algorithm

on a special class of regularized central paths for semidefinite programs we present a special class of regularized central paths for standard primal dual semidefinite program sdp that can be used to design path following algorithms for finding the projection of any symmetric matrix vector onto the solution set of the sdp we will study the existence convergence and analytical properties of these paths

a high order path following method for locating the least 2 norm solution of monotone lcps a high order path following method is proposed for finding the least 2 norm solution of a monotone lcp this method follows the regularized central paths introduced in y b zhao and d li siam j control optim 40 2001 pp 898 924 by using the analyticity of these paths we showed the global convergence under the assumption that the lcp has at least one solution and the superlinear rate of local convergence under the assumption that the least 2 norm solution is maximally complementary