lagrange multipliers in elastic plastic torsion problem for nonlinear monotone operators abstract the existence of lagrange multipliers as a radon measure is ensured for an elastic plastic torsion problem associated to a nonlinear strictly monotone operator a regularization of this result namely the existence of l p lagrange multipliers is obtained under strong monotonicity assumption on the operator moreover the relationships between elastic plastic torsion problem and the obstacle problem are investigated finally an example of the so called von mises functions is provided namely of solutions of the elastic plastic torsion problem associated to nonlinear monotone operators which are not obtained by means of the obstacle problem in the case f constant

index policies for the admission control and routing of impatient customers to heterogeneous service stations we propose a general markovian model for the optimal control of admissions and subsequent routing of customers for service provided by a collection of heterogeneous stations queue length information is available to inform all decisions admitted customers will abandon the system if required to wait too long for service the optimisation goal is the maximisation of reward rate earned from service completions net of the penalties paid whenever admission is denied and the costs incurred upon every customer loss through impatience we show that the system is indexable under mild conditions on model parameters and give an explicit construction of an index policy for admission control and routing founded on a proposal of whittle for restless bandits we are able to gain insights regarding the strength of performance of the index policy from the nature of solutions to the lagrangian relaxation used to develop the indices these insights are strengthened by the development of performance bounds although we are able to assert the optimality of the index heuristic in a range of asymptotic regimes the performance bounds are also able to identify instances where its performance is relatively weak numerical studies are used to illustrate and support the theoretical analyses

diffusion approximations for g m n gi queues with state dependent service rates we consider a sequence of many server queueing systems with impatient customers of the type g m n gi in heavy traffic this sequence is indexed by n where the parameter n represents the number of servers in the nth system the state process is considered to be the diffusion scaled total customer count in the system and the service rate is a state dependent perturbation of a given basic service rate 0 0 when the system is critically loaded in the halfin whitt heavy traffic regime we obtain the limiting diffusion for the state processes we also establish the asymptotic relationships among the diffusion scaled processes representing the total customer count virtual waiting time and the number of customer abandonments motivated by the cost structures of telephone call centers we formulate a cost functional and show that the expected value of this cost functional in the nth system converges to that of the limiting diffusion under mild assumptions

embedded trees estimation of gaussian processes on graphs with cycles graphical models provide a powerful general framework for encoding the structure of large scale estimation problems however the graphs describing typical real world phenomena contain many cycles making direct estimation procedures prohibitively costly in this paper we develop an iterative inference algorithm for general gaussian graphical models it operates by exactly solving a series of modified estimation problems on spanning trees embedded within the original cyclic graph when these subproblems are suitably chosen the algorithm converges to the correct conditional means moreover and in contrast to many other iterative methods the tree based procedures we propose can also be used to calculate exact error variances although the conditional mean iteration is effective for quite densely connected graphical models the error variance computation is most efficient for sparser graphs in this context we present a modeling example suggesting that very sparsely connected graphs with cycles may provide significant advantages relative to their tree structured counterparts thanks both to the expressive power of these models and to the efficient inference algorithms developed herein the convergence properties of the proposed tree based iterations are characterized both analytically and experimentally in addition by using the basic tree based iteration to precondition the conjugate gradient method we develop an alternative accelerated iteration that is finitely convergent simulation results are presented that demonstrate this algorithm s effectiveness on several inference problems including a prototype distributed sensing application

quasivariational solutions for first order quasilinear equations with gradient constraint we prove the existence of solutions for a quasi variational inequality of evolution with a first order quasilinear operator and a variable convex set which is characterized by a constraint on the absolute value of the gradient that depends on the solution itself the only required assumption on the nonlinearity of this constraint is its continuity and positivity the method relies on an appropriate parabolic regularization and suitable a priori estimates we also obtain the existence of stationary solutions by studying the asymptotic behaviour in time in the variational case corresponding to a constraint independent of the solution we also give uniqueness results

heavy traffic limits for queues with many exponential servers two different kinds of heavy traffic limit theorems have been proved for s server queues the first kind involves a sequence of queueing systems having a fixed number of servers with an associated sequence of traffic intensities that converges to the critical value of one from below the second kind which is often not thought of as heavy traffic involves a sequence of queueing systems in which the associated sequences of arrival rates and numbers of servers go to infinity while the service time distributions and the traffic intensities remain fixed with the traffic intensities being less than the critical value of one in each case the sequence of random variables depicting the steady state number of customers waiting or being served diverges to infinity but converges to a nondegenerate limit after appropriate normalization however in an important respect neither procedure adequately represents a typical queueing system in practice because in the heavy traffic limit an arriving customer is either a

queues with many servers the virtual waiting time process in the qed regime we consider a first come first served multiserver queue in the quality and efficiency driven qed regime in this regime which was first formalized by halfin and whitt the number of servers n is not small servers utilization is efficiency driven while waiting time is quality driven this is equivalent to having the number of servers n being approximately equal to where r is the offered load and is a positive constant for the g dk n queue in the qed regime we analyze the virtual waiting time vn t as n increases indefinitely assuming that the service time distribution has a finite support hence the dk in g dk n it is shown that in the limit the scaled virtual waiting time is representable as a supremum over a random weighted tree s denotes a service time informally it is then argued that for large n here is the averaging of over s and the process is zero mean gaussian that summarizes all relevant information about arrivals and service times arises as a limit of an infinite server g dk process appropriately scaled the results are obtained by using both combinatorial and probabilistic arguments possible applications of our approximations include fast simulation of queues and estimation prediction of customer waiting times in the qed regime

estimation in gaussian graphical models using tractable subgraphs a walk sum analysis graphical models provide a powerful formalism for statistical signal processing due to their sophisticated modeling capabilities they have found applications in a variety of fields such as computer vision image processing and distributed sensor networks in this paper we present a general class of algorithms for estimation in gaussian graphical models with arbitrary structure these algorithms involve a sequence of inference problems on tractable subgraphs over subsets of variables this framework includes parallel iterations such as embedded trees serial iterations such as block gauss seidel and hybrid versions of these iterations we also discuss a method that uses local memory at each node to overcome temporary communication failures that may arise in distributed sensor network applications we analyze these algorithms based on the recently developed walk sum interpretation of gaussian inference we describe the walks ldquocomputedrdquo by the algorithms using walk sum diagrams and show that for iterations based on a very large and flexible set of sequences of subgraphs convergence is guaranteed in walk summable models consequently we are free to choose spanning trees and subsets of variables adaptively at each iteration this leads to efficient methods for optimizing the next iteration step to achieve maximum reduction in error simulation results demonstrate that these nonstationary algorithms provide a significant speedup in convergence over traditional one tree and two tree iterations

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

robust distributed estimation using the embedded subgraphs algorithm we propose a new iterative distributed approach for linear minimum mean square error lmmse estimation in graphical models with cycles the embedded subgraphs algorithm esa decomposes a loopy graphical model into a number of linked embedded subgraphs and applies the classical parallel block jacobi iteration comprising local lmmse estimation in each subgraph involving inversion of a small matrix followed by an information exchange between neighboring nodes and subgraphs our primary application is sensor networks where the model encodes the correlation structure of the sensor measurements which are assumed to be gaussian the resulting lmmse estimation problem involves a large matrix inverse which must be solved in network with distributed computation and minimal intersensor communication by invoking the theory of asynchronous iterations we prove that esa is robust to temporary communication faults such as failing links and sleeping nodes and enjoys guaranteed convergence under relatively mild conditions simulation studies demonstrate that esa compares favorably with other recently proposed algorithms for distributed estimation simulations also indicate that energy consumption for iterative estimation increases substantially as more links fail or nodes sleep thus somewhat surprisingly sensor network energy conservation strategies such as low powered transmission and aggressive sleep schedules could actually prove counterproductive our results can be replicated using matlab code from www dsp rice edu software

dynamic scheduling of a multiclass queue in the halfin whitt heavy traffic regime we consider a markovian model of a multiclass queueing system in which a single large pool of servers attends to the various customer classes customers waiting to be served may abandon the queue and there is a cost penalty associated with such abandonments service rates abandonment rates and abandonment penalties are generally different for the different classes the problem studied is that of dynamically scheduling the various classes we consider the halfin whitt heavy traffic regime where the total arrival rate and the number of servers both become large in such a way that the system s traffic intensity parameter approaches one an approximating diffusion control problem is described and justified as a purely formal that is nonrigorous heavy traffic limit the hamilton jacobi bellman equation associated with the limiting diffusion control problem is shown to have a smooth classical solution and optimal controls are shown to have an extremal or bang bang character several useful qualitative insights are derived from the mathematical analysis including a square root rule for sizing large systems and a sharp contrast between system behavior in the halfin whitt regime versus that observed in the conventional heavy traffic regime the latter phenomenon is illustrated by means of a numerical example having two customer classes

diffusion approximations for a single node accessed by congestion controlled sources we consider simple models of congestion control in high speed networks and develop diffusion approximations which could be useful for resource allocation we first show that if the sources are on off type with exponential on and off times then under a certain scaling the steady state distribution of the number of active sources can be described by a combination of two appropriately truncated and renormalized normal distributions for the case where the source arrival process is poisson and the service times are exponential the steady state distribution consists of appropriately normalized and truncated gaussian and exponential distributions we then consider the case where the arrival process is a general renewal process with finite coefficient of variation and service time distributions that are phase type and show the impact of these distributions on the steady state distribution of the number of sources in the system we also establish an insensitivity to service time distribution when the arrival process is poisson we use these results to relate the capacity of a bottleneck node to performance measures of interest for best effort traffic

walk sums and belief propagation in gaussian graphical models we present a new framework based on walks in a graph for analysis and inference in gaussian graphical models the key idea is to decompose the correlation between each pair of variables as a sum over all walks between those variables in the graph the weight of each walk is given by a product of edgewise partial correlation coefficients this representation holds for a large class of gaussian graphical models which we call walk summable we give a precise characterization of this class of models and relate it to other classes including diagonally dominant attractive non frustrated and pairwise normalizable we provide a walk sum interpretation of gaussian belief propagation in trees and of the approximate method of loopy belief propagation in graphs with cycles the walk sum perspective leads to a better understanding of gaussian belief propagation and to stronger results for its convergence in loopy graphs

a measure type lagrange multiplier for the elastic plastic torsion abstract the existence of lagrange multiplier as a radon measure is ensured for an elastic plastic torsion problem associated to a linear operator the result is obtained by means of strong duality theory

lagrange multipliers and non constant gradient constrained problem abstract the aim of the paper is to study a gradient constrained problem associated with a linear operator two types of problems are investigated the first one is the equivalence between a non constant gradient constrained problem and a suitable obstacle problem where the obstacle solves a hamilton jacobi equation in the viscosity sense the equivalence result is obtained under a condition on the gradient constraint the second problem is the existence of lagrange multipliers we prove that the non constant gradient constrained problem admits a lagrange multiplier which is a radon measure if the free term of the equation f l p p 1 if f is a positive constant we regularize the result namely we prove that the lagrange multipliers belong to l 2

risk sensitive control for the parallel server model a markovian queueing model is considered in which servers of various types work in parallel to process jobs from a number of classes at rates mu ij that depend on the class i and the type j the problem of dynamic resource allocation so as to minimize a risk sensitive criterion is studied in a law of large numbers scaling letting x i t denote the number of class i jobs in the system at time t the cost is given by e exp n int 0 th bar x t dt g bar x t where t 0 h and g are given functions satisfying regularity and growth conditions and bar x bar x n n 1 x n cdot it is well known in an analogous context of controlled diffusion and has been shown for some classes of stochastic networks that the limit behavior as n to infty is governed by a differential game dg in which the state dynamics is given by a fluid equation for the formal limit varphi of bar x while the cost consists of int 0 th varphi t dt g varphi t and an additional term th

variational and quasivariational inequalities with first order constraints abstract we study the existence of solutions of stationary variational and quasivariational inequalities with curl constraint neumann type boundary condition and a p curl type operator these problems are studied in bounded not necessarily simply connected domains with a special geometry and the functional framework is the space of divergence free functions with curl in l p and null tangential or normal traces the analogous variational or quasivariational inequalities with gradient constraint are also studied considering neumann or dirichlet non homogeneous boundary conditions the existence of a generalized solution for a lagrange multiplier problem with homogeneous dirichlet boundary condition and the equivalence with the variational inequality is proved in the linear case for an arbitrary gradient constraint

overlapping additive and multiplicative schwarz iterations for h matrices abstract in recent years an algebraic framework was introduced for the analysis of convergence of schwarz methods for the solution of linear systems of the form ax b within this framework additive and multiplicative schwarz were shown to converge when the coefficient matrix a is a nonsingular m matrix or a symmetric positive definite matrix in this paper many of these results are extended to the case of a being an h matrix the case of inexact local solves is also considered in addition the two level scheme is studied i e when a coarse grid correction is used in conjunction with the additive or the multiplicative schwarz iterations

a new necessary and sufficient condition for the strong duality and the infinite dimensional lagrange multiplier rule abstract throughout this paper the authors introduce a new condition defined by assumption s which establishes a necessary and sufficient condition for the validity of the strong duality between a convex optimization problem and its lagrange dual this work will be focused on the context of emptiness of the interior of the ordering cone and convexity of the equality constraints moreover this new condition will be necessary and sufficient for the infinite dimensional lagrange multiplier rule this new principle will find application to the elastic plastic torsion problem to the continuum model of transportation and to a problem with quadratic equality constraint with connected to evolutionary illumination and visibility problems

gaussian markov distributions over finite graphs on s interesse aux modeles de selection de covariance introduites par dempster 1972 et etudies par wermuth 1976

a brownian control problem for a simple queueing system in the halfin whitt regime we consider a formal diffusion limit for a control problem of a multi type multi server queueing system in the regime proposed by halfin and whitt this takes the form of a control problem where the dynamics are driven by a brownian motion in one dimension a pathwise minimum is obtained and is characterized as the solution to a stochastic differential equation the pathwise solution to a special multi dimensional problem corresponding to a multi type system follows

workload dependent dynamic priority for the multiclass queue with reneging scheduling control for a single server queue with i customer classes and reneging is considered with linear holding or reneging cost an asymptotically optimal ao policy in heavy traffic is identified where classes are prioritized according to a workload dependent dynamic index rule denote by ci i and i i 1 i the queue length cost service rate and reneging rate for class i customers then a relabeling of the classes and a partition 0 w0 w1 wk k i are identified such that the policy acts to always assign least priority to the class i when the rescaled workload is in the interval wi 1 wi the relabeling is such that when workload is withing the lowest resp highest interval wi 1 wi the least priority class is the one with smallest c resp greatest value this result stands in sharp contrast to known fluid scale results where it is ao to prioritize by the fixed c index one of the technical challenges is the discontinuity of the limiting queue len

on a data driven method for staffing large call centers we consider a call center model with multiple customer classes and multiple server pools calls arrive randomly over time and the instantaneous arrival rates are allowed to vary both temporally and stochastically in an arbitrary manner the objective is to minimize the sum of personnel costs and expected abandonment penalties by selecting an appropriate staffing level for each server pool we propose a simple and computationally tractable method for solving this problem that requires as input only a few system parameters and historical call arrival data for each customer class in this sense the method is said to be data driven the efficacy of the proposed method is illustrated via numerical examples an asymptotic analysis establishes that the prescribed staffing levels achieve near optimal performance and characterizes the magnitude of the optimality gap

fair dynamic routing in large scale heterogeneous server systems in a call center there is a natural trade off between minimizing customer wait time and fairly dividing the workload among agents of different skill levels the relevant control is the routing policy that is the decision concerning which agent should handle an arriving call when more than one agent is available we formulate an optimization problem for a call center with heterogeneous agent pools in which each pool is distinguished by the speed at which agents in that pool handle calls the objective is to minimize steady state expected customer wait time subject to a fairness constraint on the workload division we first solve the optimization problem by formulating it as a markov decision process mdp and solving a related linear program we note that this approach does not in general lead to an optimal policy that has a simple structure fortunately the optimal policy does appear to have a simple structure as the system size grows large in the halfin whitt many server heavy traffic limit regime therefore we solve the diffusion control problem that arises in this regime and interpret its solution as a policy for the original system the resulting routing policy is a threshold policy that determines server pool priorities based on the total number of customers in the system we prove that a continuous modification of our proposed threshold routing policy is asymptotically optimal in the halfin whitt limit regime we furthermore present simulation results to illustrate that our proposed threshold routing policy outperforms a common routing policy used in call centers that routes to the agent that has been idle the longest

infinite dimensional duality and applications the usual duality theory cannot be applied to infinite dimensional problems because the underlying constraint set mostly has an empty interior and the constraints are possibly nonlinear in this paper we present an infinite dimensional nonlinear duality theory obtained by using new separation theorems based on the notion of quasi relative interior which in all the concrete problems considered is nonempty we apply this theory to solve the until now unsolved problem of finding in the infinite dimensional case the lagrange multipliers associated to optimization problems or to variational inequalities as an example we find the lagrange multiplier associated to a general elastic plastic torsion problem

an analysis of belief propagation on the turbo decoding graph with gaussian densities motivated by its success in decoding turbo codes we provide an analysis of the belief propagation algorithm on the turbo decoding graph with gaussian densities in this context we are able to show that under certain conditions the algorithm converges and that somewhat surprisingly though the density generated by belief propagation may differ significantly from the desired posterior density the means of these two densities coincide since computation of posterior distributions is tractable when densities are gaussian use of belief propagation in such a setting may appear unwarranted indeed our primary motivation for studying belief propagation in this context stems from a desire to enhance our understanding of the algorithm s dynamics in a non gaussian setting and to gain insights into its excellent performance in turbo codes nevertheless even when the densities are gaussian belief propagation may sometimes provide a more efficient alternative to traditional inference methods