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

lower and upper bounds for the allocation problem and other nonlinear optimization problems we demonstrate the impossibility of strongly polynomial algorithms for the allocation problem in the comparison model and in the algebraic tree computation model that follow from lower bound results consequently there are no strongly polynomial algorithms for nonlinear concave separable optimization over a totally unimodular constraint matrix this is in contrast to the case when the objective is linear we present scaling based algorithms that use a greedy algorithm as a subroutine the algorithms are polynomial for the allocation problem and its extensions and are also optimal for the sample allocation problem and the generalized upper bounds allocation problem in that the complexity meets the lower bound derived from the comparison model for other extensions of the allocation problem the scaling based algorithms presented here are the fastest known these algorithms are also polynomial time algorithms for solving with e accuracy the allocation problem and its extension in continuous variables

paths on polymatroids this paper establishes bounds on the length of certain strictly monotone paths relative to a given linear objective function on a polymatroid or on the base of a polymatroid specialized bounds are given for a strict polymatroid and for a matroid polyhedron bounds on the diameter are included

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

on structures of bisubmodular polyhedra a bisubmodular polyhedron is defined in terms of a so called bisubmodular function on a family of ordered pairs of disjoint subsets of a finite set we examine the structures of bisubmodular polyhedra in terms of signed poset and exchangeability graph we give a characterization of extreme points together with an o n 2 algorithm for discerning whether a given point is an extreme point wheren is the cardinality of the underlying set and we assume a function evaluation oracle for the bisubmodular function the algorithm also determines the signed posetructure associated with the given point if it is an extreme point we reveal the adjacency relation of extreme points by means of the hasse diagrams of the associated signed posets moreover we investigate the connectivity and the decomposition of a bisubmodular system into its connected components

the joint replenishment problem with time varying costs and demands efficient asymptotic and e optimal solutions we address the joint replenishment problem jrp where in the presence of joint setup costs dynamic lot sizing schedules need to be determined for m items over a planning horizon of n periods with general time varying cost and demand parameters we develop a new so called partitioning heuristic for this problem which partitions the complete horizon of n periods into several relatively small intervals specifies an associated joint replenishment problem for each of these and solves them via a new efficient branch and bound method the efficiency of the branch and bound method is due to the use of a new tight lower bound to evaluate the nodes of the tree a new branching rule and a new upper bound for the cost of the entire problem the partitioning heuristic can be implemented with complexity o mn2log log n it can be designed to guarantee an e optimal solution for any e 0 provided that some of the model parameters are uniformly bounded from above or below in particular the heuristic is asy

submodular functions from discrete to continous domains submodular set functions have many applications in combinatorial optimization as they can be minimized and approximately maximized in polynomial time a key element in many of the algorithms and analyses is the possibility of extending the submodular set function to a convex function which opens up tools from convex optimization submodularity goes beyond set functions and has naturally been considered for problems with multiple labels or for functions defined on continuous domains where it corresponds essentially to cross second derivatives being nonpositive in this paper we show that most results relating submodularity and convexity for set functions can be extended to all submodular functions in particular a we naturally define a continuous extension in a set of probability measures b show that the extension is convex if and only if the original function is submodular c prove that the problem of minimizing a submodular function is equivalent to a typically non smooth convex optimization problem and d propose another convex optimization problem with better computational properties e g a smooth dual problem most of these extensions from the set function situation are obtained by drawing links with the theory of multi marginal optimal transport which provides also a new interpretation of existing results for set functions we then provide practical algorithms to minimize generic submodular functions on discrete domains with associated convergence rates

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

multiclass queueing systems polymatroidal structure and optimal scheduling control in many multiclass queueing systems certain performance measures of interest satisfy strong conservation laws that is the total performance over all job types is invariant under any nonidling service control rule and the total performance over any subset say a of job types is minimized or maximized by offering absolute priority to the types in a over all other types we develop a formal definition of strong conservation laws and show that as a necessary consequence of these strong conservation laws the state space of the performance vector is a base of a polymatroid from known results in polymatroidal theory the vertices of this polyhedron are easily identified and these vertices correspond to absolute priority rules a wide variety of multiclass queueing systems are shown to have this polymatroidal structure which greatly facilitates the study of the optimal scheduling control of such systems when the defining set function of the performance space belongs to the class of generalized symmet

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

the partial order of a polymatroid extreme point given a polymatroid rank function f and its corresponding polymatroid p f we associate with each extreme point of p f a certain partial order we show that this partial order is efficiently constructible and that it characterizes all the orderings with which the greedy algorithm can be used to generate the given extreme point we give several applications including one to the still open problem of finding an efficient combinatorial procedure for testing membership in polymatroids our results can also be applied to convex games

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

the joint replenishment problem with general joint cost structures we consider inventory systems with several distinct items demands occur at constant item specific rates the items are interdependent because of jointly incurred fixed procurement costs the joint cost structure reflects general economies of scale merely assuming a monotonicity and concavity submodularity property under a power of two policy each item is replenished with constant reorder intervals which are power of two multiples of some fixed or variable base planning period our main results include a proof that depending upon whether the base planning period is fixed or variable the best among all power of two policies has an average cost which comes within either 6 or 2 of an easily computable lower bound for the minimum cost value we also derive two efficient algorithms to compute an optimal power of two policy the proposed algorithms generate as a by product a specific cost allocation of the joint cost structure to the individual items with this specific allocation the problem with separable costs is in fact equivalent to the original problem with nonseparable joint costs in the sense that the two problems share the same sets of optimal power of two policies with identical associated long run average costs

the fair resource allocation problem with submodular constraints the problem of allocating a limited resource to relevant activities in a fair manner on the basis of a certain general objective function has recently been considered by n katoh t ibaraki and h mine we generalize katoh ibaraki and mine s problem to the one with a feasible region expressed by a submodular function and give an efficient solution algorithm the proposed algorithm is of the same complexity as and simpler than katoh ibaraki and mine s when specialized to their problem

near optimal solution of generalized resource allocation problems with large capacities we consider problems of allocating resources to activities where the allocation to each activity is restricted to a general set of admissible values the objective function is additively separable but not necessarily concave nor differentiable and each activity uses at most one resource we develop a simple algorithm based on a nonsmooth convex relaxation that generates a near optimal solution whenever each allocation is a small fraction of resource capacity

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

optimization and mechanism design mechanism design is about optimizing the allocation of resources when the parameters needed to determine an optimal allocation are privately held by the agents who will consume the resources an agent s report of her information will influence the resulting allocation which in turn will affect her utility in such a situation how can one simultaneously elicit the information that is privately held and choose the optimal allocation this paper illustrates how standard results in linear programming play a role in the analysis of mechanism design problems it is not a comprehensive survey rather it focuses on two variations of a particular problem the allocation of a single object

solution techniques for some allocation problems this paper presents methods for solving allocation problems that can be stated as convex knapsack problems with generalized upper bounds such bounds may express upper limits on the total amount allocated to each of several subsets of activities in addition our model arises as a subproblem in more complex mathematical programs we therefore emphasize efficient procedures to recover optimality when minor changes in the parameters occur from one problem instance to the next these considerations lead us to propose novel data structures for such problems also we introduce an approximation method to solve certain equations which arise during the procedures

submodular optimization views on the random assignment problem we present a non pricing allocation scheme of divisible goods to agents with utility functions and submodular constraints on goods the main contribution of the present paper is that through our non pricing allocation scheme we reveal the close relation between 1 the recent results in the allocation schemes of the random assignment problem and its extensions with ordinal or lexicographic preferences on goods and 2 the monotone algorithms of fair egalitarian allocations with separable utility functions and submodular constraints investigated a few decades ago the underlying submodularity structure plays a crucial role so that the probabilistic serial mechanism of bogomolnaia and moulin and other related mechanisms can naturally be extended to problems with submodular constraints

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

optimal flows in networks with multiple sources and sinks with applications to oil and gas lease investment programs in the classical maximal flow problem the objective is to maximize the supply to a single sink in a capacitated network in this paper we consider general capacitated networks with multiple sinks the objective is to optimize a general concave preference relation on the set of feasible supply vectors we show that an optimal solution can be obtained by a marginal allocation procedure an efficient implementation results in an adaptation of the augmenting path algorithm we also discuss an application of the procedure for an investment company that deals in oil and gas ventures

a shape preserving approximation by weighted cubic splines this paper addresses new algorithms for constructing weighted cubic splines that are very effective in interpolation and approximation of sharply changing data such spline interpolations are a useful and efficient tool in computer aided design when control of tension on intervals connecting interpolation points is needed the error bounds for interpolating weighted splines are obtained a method for automatic selection of the weights is presented that permits preservation of the monotonicity and convexity of the data the weighted b spline basis is also well suited for generation of freeform curves in the same way as the usual b splines by using recurrence relations we derive weighted b splines and give a three point local approximation formula that is exact for first degree polynomials the resulting curves satisfy the convex hull property they are piecewise cubics and the curves can be locally controlled with interval tension in a computationally efficient manner

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

a decomposition algorithm for nested resource allocation problems we propose an exact polynomial algorithm for a resource allocation problem with convex costs and constraints on partial sums of resource consumptions in the presence of either continuous or integer variables no assumption of strict convexity or differentiability is needed the method solves a hierarchy of resource allocation subproblems whose solutions are used to convert constraints on sums of resources into new bounds for variables at higher levels the resulting time complexity for the integer problem is o n log m log b n and the complexity of obtaining an epsilon approximate solution for the continuous case is o n log m log b epsilon n being the number of variables m the number of ascending constraints such that m leq n epsilon a desired precision and b the total resource this algorithm matches the best known complexity when m n and improves it when log m o log n extensive experimental analyses are presented with four recent algorithms on various co

determining optimal rates for communication for omniscience this paper considers the communication for omniscience problem a set of users observe a discrete memoryless multiple source and want to recover the entire multiple source via noise free broadcast communications we study the problem of how to determine an optimal rate vector that attains omniscience with the minimum sum rate the total number of communications the results cover both asymptotic and non asymptotic models where the transmission rates are real and integral respectively we propose a modified decomposition algorithm mda and a sum rate increment algorithm sia for the asymptotic and non asymptotic models respectively both of which determine the value of the minimum sum rate and a corresponding optimal rate vector in polynomial time for the coordinate saturation capacity algorithm a nesting algorithm in mda and sia we propose to implement it by a fusion method and show by experimental results that this fusion method contributes to a reduction in computation complexity finally we show that the separable convex minimization problem over the optimal rate vector set in the asymptotic model can be decomposed by the fundamental partition the optimal partition of the user set that determines the minimum sum rate so that the problem can be solved more efficiently

optimal power of two replenishment strategies in capacitated general production distribution networks in this paper we develop a model for a capacitated production distribution network of general but acyclic topology with a general bill of materials as considered in mrp material requirement planning or drp distribution requirement planning systems this model assumes stationary deterministic demand rates and a standard stationary cost structure it is a generalization of the uncapacitated model treated in the seminal papers of maxwell and muckstadt 1985 and roundy 1986 the capacity constraints consist of bounds on the frequency with which individual items can or need to be replenished we derive a pair of simple and efficient algorithms capable of determining an optimal power of two policy these algorithms consist of a limited number of maximum flow computations in networks closely related to the production distribution network the complexity of these algorithms even when applied to the uncapacitated model compares favorably with that of the existing alternative solution methods

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

a 98 effective lot sizing rule for a multi product multi stage production inventory system we study a multi product multi stage production inventory system in continuous time in process and finished goods are referred to as products and inventories of a single item held at different locations are treated as different products external demand can occur for any or all of the products at a constant product dependent rate when an order is placed for a product it is instantly delivered the set up cost is a function of the set of products being ordered at that point in time the goal is to schedule orders for each of the products over an infinite horizon so as to minimize the long run average cost we define a new class of policies in which each product uses a stationary interval of time between successive orders and the ratio of the order intervals of any two products is an integer power of two we show that there is always a policy in this class that is within 2 of optimal this extends a similar result for the one warehouse multi retailer system for assembly systems we show how to compute such a policy in 0n log n time for general systems a sequence of max flow min cut computations is required the algorithm is efficient for very large systems

worst case analysis of greedy heuristics for integer programming with nonnegative data we give a worst case analysis for two greedy heuristics for the integer programming problem minimize cx ax b 0 x u x integer where the entries in a b and c are all nonnegative the first heuristic is for the case where the entries in a and b are integral the second only assumes the rows are scaled so that the smallest nonzero entry is at least 1 in both cases we compare the ratio of the value of the greedy solution to that of the integer optimal the error bound grows logarithmically in the maximum column sum of a for both heuristics

the line segmentation problem this paper describes a line segmentation problem in a multistage multimachine production system the production facility can concurrently produce several types of circuit boards because each production stage consists of multiple machines the items produced are categorized into families and items belonging to the same family share the common major setup while switching over from one family to another requires a major setup the line segmentation problem determines an allocation of machines at each production stage to families so as to minimize the time to complete all jobs as a result of segmenting the line several minilines are formed which are dedicated to the production of items in each family forming dedicated minilines and producing the items in a family on the same line captures the benefits of group technology and focused factory we first formalize the line segmentation problem as a quadratic integer programming problem and establish its np completeness since the problem is np complete we

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

m convex function minimization by continuous relaxation approach proximity theorem and algorithm the concept of m convexity for functions in integer variables introduced by murota adv math 124 1996 pp 272 311 plays a primary role in the theory of discrete convex analysis in this paper we consider the problem of minimizing an m convex function which is a natural generalization of the separable convex resource allocation problem under a submodular constraint and contains some classes of nonseparable convex function minimization on integer lattice points we propose a new approach for m convex function minimization based on continuous relaxation by establishing proximity theorems we develop a new algorithm based on continuous relaxation we apply the approach to some special cases of the separable convex quadratic resource allocation problem and the convex quadratic tree resource allocation problem to obtain faster algorithms

computation and efficiency of potential function minimizers of combinatorial congestion games we study the computation and efficiency of pure nash equilibria in combinatorial congestion games where the strategies of each player i are given by the binary vectors of a polytope p i our main goal is to understand which structural properties of such polytopal congestion games enable us to derive an efficient equilibrium selection procedure to compute pure nash equilibria with attractive social cost approximation guarantees to this aim we identify two general properties of the underlying aggregation polytope p n sum i p i which are sufficient for our results to go through namely the integer decomposition property idp and the box totally dual integrality property box tdi our main results for polytopal congestion games satisfying idp and box tdi are as follows i we show that pure nash equilibria can be computed in polynomial time in fact we obtain this result through a general framework for separable convex function minimization which might be of independent interest ii we bound the inefficiency of these equilibria and show that this provides a tight bound on the price of stability iii we also prove that these results extend to strong equilibria for the bottleneck variant of polytopal congestion games examples of polytopal congestion games satisfying idp and box tdi include common source network congestion games symmetric totally unimodular congestion games non symmetric matroid congestion games and symmetric matroid intersection congestion games in particular r arborescences and strongly base orderable matroids

polymatroid optimization submodularity and joint replenishment games in this paper we consider the problem of maximizing a separable concave function over a polymatroid more specifically we study the submodularity of its optimal objective value in the parameters of the objective function this question is interesting in its own right and is encountered in many applications but our research has been motivated mainly by a cooperative game associated with the well known joint replenishment model by applying our general results on polymatroid optimization we prove that this cooperative game is submodular i e its characteristic cost function is submodular if the joint setup cost is a normalized and nondecreasing submodular function furthermore the same result holds true for a more general one warehouse multiple retailer game which affirmatively answers an open question posed by anily and haviv anily s m haviv 2007 the cost allocation problem for the first order interaction joint replenishment model oper res 55 2 292 302

the discrete resource allocation problem in flow lines in this paper we address the discrete resource allocation problem in a deterministic flow line we assume that the processing times are convex and nonincreasing in the amount of resources allocated to the machines we consider the resource allocation problem for a fixed sequence of jobs for various performance criteria makespan weighted sum of completion times cycle time for cyclic schedules and develop a formulation of the problem as a convex program where the number of constraints grows exponentially with the number of jobs and machines we also present a generalization of the formulation for resource allocation problems in acyclic directed graphs we demonstrate that the problem is np complete in the strong sense and present an effective solution procedure the solution procedure is an implicit enumeration scheme where a surrogate relaxation of the formulation is used to generate upper and lower bounds on the optimal objective function value finally we address the simultaneous scheduling and resource allocation problem and we present an approximate and iterative solution procedure for the problem

coordinatewise domain scaling algorithm for m convex function minimization we present a polynomial time scaling algorithm for the minimization of an m convex function m convex functions are nonlinear discrete functions with poly matroid structures which are being recognized as playing a fundamental role in tractable cases of discrete optimization the algorithm is applicable also to a variant of quasi m convex functions

rational conformal field theory extensions of w1 in terms of bilocal fields the rational conformal field theory extensions of w1 at c 1 are in one to one correspondence with one dimensional integral lattices l m each extension is associated with a pair of oppositely charged vertex operators of charge square m n their product defines a bilocal field vm z1 z2 whose expansion in powers of z12 z1 z2 gives rise to a series of neutral local quasiprimary fields vl z m of dimension l 1 the associated bilocal exponential of a normalized current generates the w1 algebra spanned by the vl z 1 and the unit operator the extension of this construction to higher integer values of the central charge c is also considered applications to a quantum hall system require computing characters i e chiral partition functions depending not just on the modular parameter but also on a chemical potential we compute such a dependence of orbifold characters thus extending the range of applications of a recent study of affine orbifolds

parallel mesh methods for tension splines this paper addresses the problem of shape preserving spline interpolation formulated as a differential multipoint boundary value problem dmbvp for short its discretization by mesh method yields a five diagonal linear system which can be ill conditioned for unequally spaced data using the superposition principle we split this system in a set of tridiagonal linear systems with a diagonal dominance the latter ones can be stably solved either by direct gaussian elimination or iterative methods sor method and finite difference schemes in fractional steps and admit effective parallelization numerical examples illustrate the main features of this approach

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

sharp error bounds for interpolating splines in tension abstract sharp error bounds for interpolating splines in tension with variable tension parameters are considered error bounds given are the best possible ones in the limit case of the tension parameter p that is for linear p and cubic p 0 spline error bounds sharper than those previously published are also developed for derivatives

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

establishing consistent and realistic reorder intervals in production distribution systems the objective of this paper is to present a model and algorithm that can be used to find consistent and realistic reorder intervals for each item in large scale production distribution systems we assume such systems can be represented by directed acyclic graphs demand for each end item is assumed to occur at a constant and continuous rate production is instantaneous and no backorders are allowed both fixed setup costs and echelon holding costs are charged at each stage we limit our attention to nested and stationary policies furthermore we restrict the reorder interval for each stage to be a power of 2 times a base planning period the model that results from these assumptions is an integer nonlinear programming problem the optimal solution can be found using the proposed algorithm which is a polynomial time algorithm a real world example is given to illustrate the procedure

least majorized elements and generalized polymatroids we prove that a bounded generalized polymatroid has a least weakly submajorized supermajorized vector such a vector simultaneously minimizes every nondecreasing nonincreasing symmetric and quasi convex function defined on the generalized polymatroid the same result holds also for the set of integer vectors of a bounded integral generalized polymatroid we then extend these results to more general sets and discuss several computational aspects

a critical review of discrete filled function methods in solving nonlinear discrete optimization problems many real life problems can be modeled as nonlinear discrete optimization problems such problems often have multiple local minima and thus require global optimization methods due to high complexity of these problems heuristic based global optimization techniques are usually required when solving large scale discrete optimization or mixed discrete optimization problems one of the more recent global optimization tools is known as the discrete filled function method nine variations of the discrete filled function method in literature are identified and a review on theoretical properties of each method is given some of the most promising filled functions are tested on various benchmark problems numerical results are given for comparison

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

a combinatorial strongly polynomial algorithm for minimizing submodular functions this paper presents a combinatorial polynomial time algorithm for minimizing submodular functions answering an open question posed in 1981 by grotschel lovasz and schrijver the algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter the resulting algorithm runs in time bounded by a polynomial in the size of the underlying set and the length of the largest absolute function value the paper also presents a strongly polynomial version in which the number of steps is bounded by a polynomial in the size of the underlying set independent of the function values

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

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

generalized polymatroids and submodular flows polyhedra related to matroids and sub or supermodular functions play a central role in combinatorial optimization the purpose of this paper is to present a unified treatment of the subject the structure of generalized polymatroids and submodular flow systems is discussed in detail along with their close interrelation in addition to providing several applications we summarize many known results within this general framework

classification of the local extensions of the su 3 chiral current algebra all local extensions of the chiral algebra of observables in the su 2 r conformal current algebra models are classified for r 4 and all levels ki this method is based on analyzing the polynomial solutions of the corresponding knizhnik zamolodchikov equations and can be applied also for explicit construction of all local extensions corresponding to conformal embeddings of su 2 r for any r

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

on resource allocation in fading multiple access channels an efficient approximate projection approach in this paper we consider the problem of rate and power allocation in a multiple access channel mac our objective is to obtain rate and power allocation policies that maximize a general concave utility function of average transmission rates on the information theoretic capacity region of the mac without using queue length information first we address the utility maximization problem in a nonfading channel and present a gradient projection algorithm with approximate projections by exploiting the polymatroid structure of the capacity region we show that the approximate projection can be implemented in time polynomial in the number of users second we present optimal rate and power allocation policies in a fading channel where channel statistics are known for the case that channel statistics are unknown and the transmission power is fixed we propose a greedy rate allocation policy and characterize the performance difference of this policy and the optimal policy in terms of channel variations and structure of the utility function the numerical results demonstrate superior convergence rate performance for the greedy policy compared to queue length based policies in order to reduce the computational complexity of the greedy policy we present approximate rate allocation policies which track the greedy policy within a certain neighborhood

using separable programming to solve the multi product multiple ex ante constraint newsvendor problem and extensions this paper provides an approximating programming technique to solve the multi product newsvendor model in which product demands are independent and stocking quantities are subject to two or more ex ante linear contraints such as budget or volume constraints previous research has attempted to solve this problem with lagrange relaxation techniques or by limiting the distribution of demand however by taking advantage of the separable nature of the problem a close approximation of the optimal solution can be found using convex separable programming for any demand distribution in the traditional newsvendor model and extensions sensitivity analysis of the linear program provides managerial insight into the effects of parameters of the problem on the optimal solution and future decisions

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

quasifinite highest weight modules over the lie algebra of differential operators on the circle we classify positive energy representations with finite degeneracies of the lie algebraw 1 and construct them in terms of representation theory of the lie algebra hat gl infty r m of infinites matrices with finite number of non zero diagonals over the algebrar m t t m 1 the unitary ones are classified as well similar results are obtained for the sin algebras

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

layered augmenting path algorithms augmenting path algorithms first introduced by ford and fulkerson ford l r d r fulkerson 1962 flows in networks princeton university press princeton n j are widely used in optimization examples include schonsleben s polymatroid intersection algorithm schonsleben p 1980 ganzzahlige polymatroid intersection algorithmen ph d thesis eidgenossischen technischen hochschule zurich the maximum polymatroidal network flow algorithm of lawler and martel lawler e l c u martel 1982 computing maximal polymatroidal network flows math oper res 7 334 347 frank s algorithm for the edmonds giles polyhedron frank a 1984 finding feasible vectors of edmonds giles polyhedra j combin theory ser b36 221 239 and cunningham s algorithm for testing membership in matroid polyhedra cunningham h w 1984 testing membership in matroid polyhedra j combin theory ser b36 161 188 here we give an order of magnitude improvement for the above algorithms by using an approach analogous to that of dinits maximum flow algorithm dinits e a 1970 algorithm for solution of a problem of maximum flow in a network with power estimation soviet math dokl 11 1277 1280

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