adaptive lexicographic optimization in multi class m gi 1 queues we consider a multi class m gi 1 system in which an average response time objective is associated with each class the performance of each class is measured by the ratio of the average response time over the corresponding value of the objective to achieve fairness in service allocation it is required to find a policy that lexicographically minimizes the vector of performance ratios arranged in nonincreasing order we provide such a policy that is adaptive uses only knowledge of arrival and departure instants and is thus easy to implement we also consider a variant of this policy which adapts faster to changes in the statistical parameters of the model both policies are analyzed via associated stochastic recursions using techniques of stochastic approximation

large scale tikhonov regularization via reduction by orthogonal projection abstract this paper presents a new approach to computing an approximate solution of tikhonov regularized large scale ill posed least squares problems with a general regularization matrix the iterative method applies a sequence of projections onto generalized krylov subspaces a suitable value of the regularization parameter is determined by the discrepancy principle

restless bandit marginal productivity indices diminishing returns and optimal control of make to order make to stock m g 1 queues this paper presents a framework grounded on convex optimization and economics ideas to solve by index policies problems of optimal dynamic allocation of effort to a discrete state finite or countable binary action work rest semi markov restless bandit project elucidating issues raised by previous work its contributions include i the concept of a restless bandits marginal productivity index mpi characterizing optimal policies relative to general cost and work measures ii the characterization of indexable restless bandits as those satisfying diminishing marginal returns to work consistently with a nested family of threshold policies iii sufficient indexability conditions via partial conservation laws pcls iv the characterization of the mpi as an optimal marginal productivity rate relative to feasible active state sets v application to semi markov bandits under several criteria including a new mixed average bias criterion and vi pcl indexability analyses and mpis for optimal service control of make to order make to stock queues with convex holding costs under discounted and average bias criteria

a global lanczos method for image restoration image restoration often requires the solution of large linear systems of equations with a very ill conditioned possibly singular matrix and an error contaminated right hand side the latter represents the available blur and noise contaminated image while the matrix models the blurring computation of a meaningful restoration of the available image requires the use of a regularization method we consider the situation when the blurring matrix has a kronecker product structure and an estimate of the norm of the desired image is available and illustrate that efficient restoration of the available image can be achieved by tikhonov regularization based on the global lanczos method and by using the connection of the latter to gauss type quadrature rules

problems of adaptive optimization in multiclass m gi 1 queues with bernoulli feedback adaptive algorithms are obtained for the solution of separable optimization problems in multiclass m gi 1 queues with bernoulli feedback optimality of the algorithms is established by modifying and extending methods of stochastic approximation these algorithms can be used as a basis for designing policies for semi separable and approximate lexicographic optimization problems and in the case of m gi 1 queues without feedback they also provide a simple policy for lexicographic optimization the results obtained on stochastic approximation imply convergence of classical recursions such as robbins monroe in cases where the conditional second moment of their increments is not finite

amenable and weakly amenable banach algebras with compact multiplication abstract we investigate amenable and weakly amenable banach algebras with compact multiplication any amenable banach algebra with compact multiplication is biprojective as a consequence every semisimple such algebra which has the approximation property is a topological direct sum of full matrix algebras in the radical case no such structure theorem is at hand we also investigate banach algebras which have a bounded approximate identity consisting of normalized powers of an element x any such banach algebra is either unital or radical if the algebra is also generated by x it is weakly amenable we construct a radical example with compact multiplication which moreover is an integral domain this furnishes a new example of a commutative weakly amenable non amenable radical banach algebra

submodular function minimization submodular functions often arise in various fields of operations research including discrete optimization game theory queueing theory and information theory in this survey paper we give overview on the fundamental properties of submodular functions and recent algorithmic devolopments of their minimization

the greedy procedure for resource allocation problems necessary and sufficient conditions for optimality in many resource allocation problems the objective is to allocate discrete resource units to a set of activities so as to maximize a concave objective function subject to upper bounds on the total amounts allotted to certain groups of activities if the constraints determine a polymatroid and the objective is linear it is well known that the greedy procedure results in an optimal solution in this paper we extend this result to objectives that are weakly concave a property generalizing separable concavity we exhibit large classes of models for which the set of feasible solutions is a polymatroid and for which efficient implementations of the greedy procedure can be given

decomposable markov decision processes a fluid optimization approach decomposable markov decision processes mdps are problems where the stochastic system can be decomposed into multiple individual components although such mdps arise naturally in many practical applications they are often difficult to solve exactly due to the enormous size of the state space of the complete system which grows exponentially with the number of components in this paper we propose an approximate solution approach to decomposable mdps that is based on re solving a fluid linear optimization formulation of the problem at each decision epoch this formulation tractably approximates the problem by modeling transition behavior at the level of the individual components rather than the complete system we prove that our fluid formulation provides a tighter bound on the optimal value function than three state of the art formulations the approximate linear optimization formulation the classical lagrangian relaxation formulation and a novel alternate lagrangian relaxation that is based on relax

global golubkahan bidiagonalization applied to large discrete ill posed problems we consider the solution of large linear systems of equations that arise from the discretization of ill posed problems the matrix has a kronecker product structure and the right hand side is contaminated by measurement error problems of this kind arise for instance from the discretization of fredholm integral equations of the first kind in two space dimensions with a separable kernel and in image restoration problems regularization methods such as tikhonov regularization have to be employed to reduce the propagation of the error in the right hand side into the computed solution we investigate the use of the global golubkahan bidiagonalization method to reduce the given large problem to a small one the small problem is solved by employing tikhonov regularization a regularization parameter determines the amount of regularization the connection between global golubkahan bidiagonalization and gauss type quadrature rules is exploited to inexpensively compute bounds that are useful for determining the regularization parameter by the discrepancy principle

characterization and optimization of achievable performance in general queueing systems this paper considers general single facility queueing systems with exponential service times dealing with a finite number j of distinct customer classes performance of the system as measured by the vector of steady state expected sojourn times of the customer classes the performance vector may be controlled by adopting an appropriate preemptive priority discipline we show that the performance space the set of performance vectors which are achievable under some preemptive work conserving rule is a polyhedron described by 2j 1 inequalities the special structure of this polyhedron nevertheless allows for efficient procedures to minimize any separable convex function of the performance vector linear objectives are shown to be minimized by absolute priority rules thus generalizing a well known result for m m 1 systems we also show that each point in the performance space may be achieved by a specific randomization of at most j 1 absolute priority rules

a generalized matrix krylov subspace method for tv regularization abstract this paper presents efficient algorithms to solve both tv l1 and tv l2 models of images contaminated by blur and noise the unconstrained structure of the problems suggests that one can solve a constrained optimization problem by transforming the original unconstrained minimization problem to an equivalent constrained minimization one an augmented lagrangian method is developed to handle the constraints when the model is given with matrix variables and an alternating direction method adm is used to iteratively find solutions of the subproblems the solutions of some subproblems are belonging to subspaces generated by application of successive orthogonal projections onto a class of generalized matrix krylov subspaces of increasing dimension we give some theoretical results and report some numerical experiments to show the effectiveness of the proposed algorithms

convergent expansions and bounded operators in the umbral calculus on etudie des proprietes analytiques de certaines suites de fonctions speciales appelees suites de sheffer

adjacency on polymatroids this paper characterizes adjacency for extreme points of a polymatroid two extreme points of a polymatroid are adjacent if and only if they either differ in exactly one component or differ in exactly two components with the two components satisfying a certain ordering relation a polynomial algorithm generates and lists all extreme points adjacent to a given extreme point of a polymatroid similar results hold for the core of a convex game

multiclass queueing systems in heavy traffic an asymptotic approach based on distributional and conservation laws we propose a new approach to analyze multiclass queueing systems in heavy traffic based on what we consider as fundamental laws in queueing systems namely distributional and conservation laws methodologically we extend the distributional laws from single class queueing systems to multiple classes and combine them with conservation laws to find the heavy traffic behavior of the following systems a gi g 1 queue under fifo b gi g 1 queue with priorities c polling systems with general arrival distributions compared with traditional heavy traffic analysis via brownian processes our approach gives new insight to the asymptotics used solves systems that traditional heavy traffic theory has not fully addressed and more importantly leads to closed form answers which compared to simulation are very accurate even for moderate traffic

matrices moments and rational quadrature abstract many problems in science and engineering require the evaluation of functionals of the form f u a u t f a u where a is a large symmetric matrix u a vector and f a nonlinear function a popular and fairly inexpensive approach to determining upper and lower bounds for such functionals is based on first carrying out a few steps of the lanczos procedure applied to a with initial vector u and then evaluating pairs of gauss and gauss radau quadrature rules associated with the tridiagonal matrix determined by the lanczos procedure the present paper extends this approach to allow the use of rational gauss quadrature rules

computing maximal polymatroidal network flows in the classical network flow model flows are constrained by the capacities of individual arcs in the polymatroidal network flow model introduced in this paper flows are constrained by the capacities of sets of arcs yet the essential features of the classical model are retained the augmenting path theorem the integral flow theorem and the max flow min cut theorem arc all shown to yield to straightforward generalization we describe a maximal flow algorithm which finds augmenting paths by labeling arcs instead of nodes as in the case of the classical model as a counterpart of a known result for the classical model we prove that the number of augmentations required to achieve a maximal value flow is bounded by the cube of the number of arcs in the network provided each successive augmentation is made along a shortest augmenting path with ties between shortest paths broken by lexicography

two algorithms for maximizing a separable concave function over a polymatroid feasible region abstract in this paper we present two new algorithms for maximizing a separable concave function on a polymatroid both algorithms apply to the discrete as well as the continuous version of the problem the application of these algorithms to several types of polymatroids is discussed and we show that the decomposition algorithm runs in polynomial time in the discrete version for network and generalized symmetric polymatroids and the bottom up algorithm in the discrete version runs in polynomial time when the polymatroid is given as an explicit list of constraints

restless bandits linear programming relaxations and a primal dual index heuristic we develop a mathematical programming approach for the classicalpspace hard restless bandit problem in stochastic optimization we introduce a hierarchy ofn wheren is the number of bandits increasingly stronger linear programming relaxations the last of which is exact and corresponds to the exponential size formulation of the problem as a markov decision chain while the other relaxations provide bounds and are efficiently computed we also propose a priority index heuristic scheduling policy from the solution to the firstorder relaxation where the indices are defined in terms of optimal dual variables in this way we propose a policy and a suboptimality guarantee we report results of computational experiments that suggest that the proposed heuristic policy is nearly optimal moreover the second order relaxation is found to provide strong bounds on the optimal value

dynamic allocation indices for restless projects and queueing admission control a polyhedral approach this paper develops a polyhedral approach to the design analysis and computation of dynamic allocation indices for scheduling binary action engage rest markovian stochastic projects which can change state when rested restless bandits rbs based on partial conservation laws pcls this extends previous work by the author j nino mora 2001 restless bandits partial conservation laws and indexability adv appl probab 33 76 98 where pcls were shown to imply the optimality of index policies with a postulated structure in stochastic scheduling problems under admissible linear objectives and they were deployed to obtain simple sufficient conditions for the existence of whittle s 1988 rb index indexability along with an adaptive greedy index algorithm the new contributions include i we develop the polyhedral foundation of the pcl framework based on the structural and algorithmic properties of a new polytope associated with an accessible set system extended polymatroid ii we present new dynamic allocation indices for rbs motivated by an admission control model which extend whittle s and have a significantly increased scope iii we deploy pcls to obtain both sufficient conditions for the existence of the new indices pcl indexability and a new adaptive greedy index algorithm iv we interpret pcl indexability as a form of the classic economics law of diminishing marginal returns and characterize the index as an optimal marginal cost rate we further solve a related optimal constrained control problem v we carry out a pcl indexability analysis of the motivating admission control model under time discounted and long run average criteria this gives under mild conditions a new index characterization of optimal threshold policies and vi we apply the latter to present new heuristic index policies for two hard queueing control problems admission control and routing to parallel queues and scheduling a multiclass make to stock queue with lost sales both under state dependent holding cost rates and birth death dynamics

continuous sheffer families ii abstract continuous sheffer families have been recently introduced by the authors these are continuous versions of the sheffer sequences arising in the umbral calculus we show here that quite a number of classical special functions are examples of such families

banach algebras logarithms and polynomials of convolution type abstract using elementary banach algebra techniques it is determined which elements of banach algebras like l1 b k and a have logarithms the wiener levy theorem will be used to answer the same question for more complicated banach algebras like the wiener algebra these results will be applied to polynomials of convolution type and a generalization thereof

branching bandits and klimov s problem achievable region and side constraints we consider the average cost branching bandits problem and its special case known as klimov s problem we consider the vector n whose components are the mean number of bandits or customers of each type that are present we characterize fully the achievable region that is the set of all possible vectors n that can be obtained by considering all possible policies while the original description of the achievable region involves exponentially many constraints we also develop an alternative description that involves only o r sup 2 variables and constraints where r is the number of bandit types or customer classes we then consider the problem of minimizing a linear function of n subject to l additional linear constraints on n we show that optimal policies can be obtained by randomizing between l 1 strict priority policies that can be found efficiently in polynomial time using linear programming techniques

index policies and a novel performance space structure for a class of generalised branching bandit problems our object of study is a new class of controlled stochastic systems called generalised branching bandits which include discounted branching bandits and generalised bandit problems as special cases these models allow us to study queueing scheduling and project scheduling problems in which the reward earned from processing a particular job is influenced by other job types present in the system our mode of analysis is the achievable region approach what is novel here is that the full system may be partitioned into two subsystems each of which satisfies its own set of conservation laws and has its own highly structured performance space an optimal policy for the full system is constructed from two sets of gittins indices derived from the conservation laws governing the two subsystems

submodular returns and greedy heuristics for queueing scheduling problems we consider a range of controlled stochastic systems satisfying conservation laws together with a reducibility property that says that appropriate laws continue to hold when access to the system is restricted to a subset of all possible demand job customer types we show that for linear objectives the optimum system wide performance is a nondecreasing submodular or supermodular function of the subset chosen and that these properties are inherited from the geometry of the performance space concerned these results are of considerable interest in their own right but they also motivate the use of greedy heuristics for the solution of a range of job selection and scheduling problems which have hitherto seemed intractable computational experience suggests that such heuristics perform very well

existence of limiting distributions in the gi g s queue a new proof is given for the kiefer wolfowitz kiefer j j wolfowitz 1955 on the theory of queues with many servers trans amer math soc 78 1 18 theorem in the gi g s queue the sequence of workload vectors and thus also the sequence of waiting times has a proper limiting distribution independent of the initial conditions if and only if p 0 1 assuming the interarrival times and service times have finite means from stochastic monotonicily convergence is easy to establish if the system starts off empty the limit is shown lo be proper here by bounding the standard fcfs system by the corresponding s server system with a cyclic assignment discipline using a recent construction of wolff wolff r w 1977 an upper bound for multi channel queues j appl probab 14 884 888 the general initial condition case is treated by appropriately bounding the original sequence above and below by sequences of countable state markov chains for which zero is always a regenerative state this paper thus illustrates how recent stochastic comparison and continuity results can be applied to prove limit theorems the bounding systems are also of special interest in regenerative simulation because they provide a way lo increase the number of regeneration points

kronecker product and svd approximations in image restoration abstract image restoration applications often result in ill posed least squares problems involving large structured matrices one approach used extensively is to restore the image in the frequency domain thus providing fast algorithms using fast fourier transforms ffts this is equivalent to using a circulant approximation to a given matrix iterative methods may also be used effectively by exploiting the structure of the matrix while iterative schemes are more expensive than fft based methods it has been demonstrated that they are capable of providing better restorations as an alternative we propose an approximate singular value decomposition svd which can be used in a variety of applications used as a direct method the computed restorations are comparable to iterative methods but are computationally less expensive in addition the approximate svd may be used with the generalized cross validation method to choose regularization parameters it is also demonstrated that the approximate svd can be an effective preconditioner for iterative methods

efficient algorithms for finding optimal power of two policies for production distribution systems with general joint setup costs we consider a production distribution system represented by a general directed acyclic network each node is associated with a specific product at a given location and or production stage an arc i j indicates that item i is used to produce item j external demands may occur at any of the network s nodes these demands occur continuously at item specific constant rates components may be assembled in any given proportions the cost structure consists of inventory carrying variable and fixed production distribution costs the latter depend at any given replenishment epoch on the specific set of items being replenished according to an arbitrary set function merely assumed to be monotone and submodular it has been shown that a simply structured so called power of two policy is guaranteed to come within 2 of a lower bound for the minimum cost in this paper we derive efficient algorithms for the computation of an optimal power of two policy possibly in combination with this lower bound these consist of a limited number of polymatroidal maximum flow calculations in networks closely associated with the original production distribution network

m g c queueing systems with multiple customer classes characterization and control of achievable performance under nonpreemptive priority rules this paper considers an m g c queueing system serving a finite number j of distinct customer classes performance of the system as measured by the vector of steady state expected waiting times of the customer classes the performance vector may be controlled by adopting an appropriate priority discipline we show that the performance space the set of performance vectors which are achievable under some nonpreemptive work conserving priority rule is a polyhedron described by 2j 1 inequalities the special polymatroidal structure of this polyhedron nevertheless allows for efficient oj2 log j procedures to minimize any convex separable function of the performance vector linear objectives are shown to be minimized by absolute priority rules thus generalizing a well known result for m g 1 systems we also show that each point in the performance space may be achieved by a unique generalized dynamic priority rule specified by j 1 parameters which may be determined by the recursive solution of j 1 single variable quadratic equations this class of rules contains the absolute priority rules and the pure dynamic rules as special cases our results are accurate up to one extremely accurate approximation and completely exact for m g 1 and m m c systems as well as in heavy traffic

stochastic scheduling with priority classes we consider controlled stochastic systems in which multiple job types customers projects jobs etc may require processing by one or more servers the job types are all members of priority classes and admissible controls must respect the constraints imposed by these for systems which when unconstrained satisfy generalised conservation laws we obtain the performance space and show that linear objectives are optimised by admissible controls which choose job types within each priority class according to a set of gittins indices various degrees of decomposition of the problem are described and illustrated in the special case of discounted and undiscounted branching bandits the theory is developed further to yield index based suboptimality bounds for general policies

conditions for finite ladder height and delay moments we provide simple proofs for several known results conditions for finite ladder height moments are obtained in a unified way for random walks under both the oscillating and drift cases the random walk conditions are used to obtain conditions for finite delay moments for the gi g 1 queue the gi g 1 conditions are then used to obtain the kiefer and wolfowitz conditions for finite moments of delay and of work in system for the gi g c queue

sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis abstract for certain sheffer sequences s n n 0 on c grabiner 1988 proved that for each 0 1 the corresponding sheffer operator z n s n z extends to a linear self homeomorphism of e min c the frechet topological space of entire functions of order at most and minimal type when the order is equal to 0 in particular every function f e min c admits a unique decomposition f z n 0 c n s n z and the series converges in the topology of e min c within the context of a complex nuclear space and its dual space in this work we generalize grabiner s result to the case of sheffer operators corresponding to sheffer sequences on in particular for c n with n 2 we obtain the multivariate extension of grabiner s theorem furthermore for an appell sequence on a general co nuclear space we find a sufficient condition for the corresponding sheffer operator to extend to a linear self homeomorphism of e min when 1 the latter result is new even in the one dimensional case

on the conservation law and the performance space of single server systems we consider a multiclass gi g 1 queueing system operating under an arbitrary work conserving scheduling policy we derive an invariance relation for the cesaro sums of waiting times under which does not require the existence of limits of the cesaro sums this allows us to include important classes in the set of admissible policies such as time dependent and adaptive policies for these classes of policies ergodicity is not known a priori and may not even exist therefore the classical invariance relations that involve statistical averages do not hold for an m g 1 system we derive inequalities involving the cesaro sums of waiting times that further characterize the achievable performance region of the system

global least squares method gl lsqr for solving general linear systems with several right hand sides abstract in this paper we propose a new method for solving general linear systems with several right hand sides this method is based on global least squares method and reduces the original matrix to the lower bidiagonal form we derive a simple recurrence formula for generating the sequence of approximate solutions x k some theoretical properties of the new method are discussed and we also show that how this method can be implemented for the sylvester equation finally some numerical experiments on test matrices are presented to show the efficiency of the new method