efficient algorithms for the inverse spanning tree problem the inverse spanning tree problem is to modify edge weights in a graph so that a given treet is a minimum spanning tree the objective is to minimize the cost of the deviation the cost of deviation is typically a convex function we propose algorithms here that are faster than all known algorithms for the problem our algorithm s run time for any convex inverse spanning tree problem iso nm log 2n for a graph onn nodes andm edges plus the time required to find the minima of then convex deviation functions this not only improves on the complexity of previously known algorithms for the general problem but also for algorithms devised for special cases for the case when the objective is weighted absolute deviation sokkalingam et al 1999 devised an algorithm with run timeo n2m log nc forc the maximum edge weight for the sum of absolute deviations our algorithm runs in timeo n2 logn matching the run time of a recent ahuja and orlin 2000 improvement for this case a new algorithm for bipartite matching in path graphs is reported here with complexity ofo n1 5 logn this leads to a second algorithm for the sum of absolute deviations running ino n1 5 logn logc steps

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

dynamic recomputation cannot extend the optimality range of priority indices sequencing rules that rely on priority indices are particularly attractive because they are simple to calculate and easy to implement it has been established that there are exactly two classes of delay cost functions for which policies that are determined by time invariant priority indices are capable of producing optimal sequences linear delay costs and discounted linear delay costs we consider index based policies that allow dynamic recalculation of priority indices and show that this class does not enlarge the set of delay cost functions for which index based rules can produce optimal policies our analysis relies on the argument that index based rules must induce a transitive ranking of the tasks we show that the classes of linear and discounted linear functions are the only ones that can be associated with such rankings

a demand to manufacturing requirement model based on predefined configurations abstract this paper examines the production planning of a job shop manufacturing system capable of producing various part types the demand for the various product types is known at the end of each elementary period the objective function is the sum of inventory and backlog costs accumulated during a finite period of time known as the horizon of the problem this horizon is a multiple of the length of an elementary period we define the new concept of configuration as a set of maximal quantities of each type of parts which can be manufactured during an elementary period of time this set is chosen either in order to balance the resource loading or to optimize the use of the tools available in the job shop we assume that there exists a finite set of configurations the changeover of the system from one configuration to another is only allowed at the end of an elementary period and introduces a set up time during which no machine can produce the control of such a system consists in choosing the sequence of configurations and the actual production levels on each elementary period so as to minimize the objective function we propose an efficient optimal algorithm for computing the actual production levels on each elementary period when the sequence of configurations i e the sequence of constraints is given a heuristic algorithm is also proposed to find a good sequence of configurations

monotonicity recovering and accuracy preserving optimization methods for postprocessing finite element solutions we suggest here a least change correction to available finite element fe solution this postprocessing procedure is aimed at recovering the monotonicity and some other important properties that may not be exhibited by the fe solution although our approach is presented for fes it admits natural extension to other numerical schemes such as finite differences and finite volumes for the postprocessing a priori information about the monotonicity is assumed to be available either for the whole domain or for a subdomain where the lost monotonicity is to be recovered the obvious requirement is that such information is to be obtained without involving the exact solution e g from expected symmetries of this solution the postprocessing is based on solving a monotonic regression problem with some extra constraints one of them is a linear equality type constraint that models the conservativity requirement the other ones are box type constraints and they originate from the discrete maximum principle the resulting postprocessing problem is a large scale quadratic optimization problem it is proved that the postprocessed fe solution preserves the accuracy of the discrete fe approximation we introduce an algorithm for solving the postprocessing problem it can be viewed as a dual ascent method based on the lagrangian relaxation of the equality constraint we justify theoretically its correctness its efficiency is demonstrated by the presented results of numerical experiments

note on the optimality of integer lot size ratios in economic lot size determination in multi stage assembly systems the proof of a well known theorem by crowston and wagner in crowston wagner and williams crowston w b m h wagner j f williams 1973 economic lot size determination in multi stage assembly systems management sci 19 5 january 517 527 is shown here to be defective according to this theorem an optimal solution to the batch size determination problem for multi echelon production inventory assembly structures is characterized by a set of lot sizes such that the lot size at each stage must be an integer multiple of the lot size at its successor stage the theorem has been said to apply to uncapacitated assembly systems having constant final product demand over an infinite horizon with no backlogs or lost sales permitted where lot sizes must be rational numbers and time invariant while the theorem supposedly applied to the case of both finite production rates and infinite production rates the proof dealt only with the case of instantaneous production also the proof implicitly assumed that lots were processed at any given stage only after unchanging spans of time in this paper we show that with or without this implicit assumption the theorem does hold true for the special case of instantaneous processing in two level assembly systems that is configurations where there is only one successor stage in the entire system however we also show by counterexample that without this implicit assumption the theorem is invalid for more general assembly structures furthermore in the appendix of this paper we show that the proof is defective at the point where crowston and wagner extend their results for two level systems to more general assembly systems thus it is an open question whether or not the theorem is valid for all assembly structures when processing at any given stage must occur only after unchanging spans of time

dynamic lot sizing for multi echelon distribution systems with purchasing and transportation price discounts we consider the problem of determining optimal purchasing and shipping quantities over a finite planning horizon for arborescent multi echelon physical distribution systems with deterministic time varying demands we assume that the inventory holding cost at a given warehouse of the distribution network is a linear function of the inventory level and that the total procurement cost i e ordering plus purchasing plus transportation and reception costs is a general piecewise linear function of the quantities shipped to and from the warehouse we formulate a mixed integer linear programming model of the problem and develop a lagrangian relaxation based procedure to solve it we show computational results for problems with 12 periods up to 15 warehouses and 3 transportation price ranges

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

a finite method for the solution of a multi resource allocation problem with concave return functions it has recently been recognized that the problem of allocating continuous resources to several activities each with a concave return function has some elements in common with the transportation problem this has inspired a search for solution algorithms for the allocation problem in which the approach used is similar to that applied to the transportation problem this paper describes such an algorithm the algorithm terminates and produces the exact optimal solution a fortran implementation of the algorithm solves a 12 resources 100 activities problem in less than 10 central processor seconds on a cyber 173 computer

a fast scaling algorithm for minimizing separable convex functions subject to chain constraints we consider the problem of minimizing j nc j x j subject to the following chain constraints x 1 x 2 x 3 x n where c j x j is a convex function of x j for each j n 1 2 n this problem is a generalization of theisotonic regression problems with complete order an important class of problems in regression analysis that has been examined extensively in the literature we refer to this problem as thegeneralized isotonic regression problem in this paper we focus on developing a fast scaling algorithm to obtain an integer solution of the generalized isotonic regression problem letu denote the difference between an upper bound on an optimal value of x n and a lower bound on an optimal value of x 1 under the assumption that the evaluation of any function c j x j takeso 1 time we show that the generalized isotonic regression problem can be solved ino n logu time this improves by a factor of n the previous best running time ofo n 2 logu to solve the same problem in addition when our algorithm is specialized to theisotonic median regression problem where c j x j c j x j a ji for specified values ofc js anda js the algorithm obtains a real valued optimal solution ino n logn time this time bound matches the best available time bound to solve the isotonic median regression problem but our algorithm uses simpler data structures and may be easier to implement

absolute continuity of the best sobolev constant abstract let q inf u l p p u l q p u w 0 1 p 0 be the best sobolev constant of the immersion w 0 1 p l q where is a bounded and smooth domain of r n 1 p n and 1 q p n p n p we prove that the function q 1 p q is absolutely continuous

simple power of two policies are close to optimal in a general class of production distribution networks with general joint setup costs we consider a production distribution network represented by a general directed acyclic network each node is associated with a specific product or item 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 and 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 we show that a simply structured so called power of two policy is guaranteed to come within 2 of a lower bound for the minimum cost under a power of two policy all items are replenished at constant intervals and only when their inventory drops to zero moreover these replenishment intervals are all power of two multiples of a common base planning period the above results generalize those of roundy 1986

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

design of a 3 3 multivariable feedback system with large plant uncertainty a recent paper presented a quantitative synthesis theory based on schauder s fixed point theorem for uncertain multiple input output feedback systems this paper is devoted to practical design execution by means of a detailed 3 3 problem with large plant uncertainty including open loop instability and some non minimum phase plant elements the outstanding features of the theory are 1 the quantitative nature of the assigned tolerances on the nine closed loop system response functions in this problem four are specified basically non interacting and the other five interacting 2 the design problem is translated into separate quantitative single loop problems each with plant uncertainty external disturbances and closed loop tolerances derived from the original problem the solutions for these single loop problems are guaranteed to solve the original multivariable problem whose system characteristic equation need never be examined 3 there is developed a systematic means of a optimization of the

heuristics for multilevel lot sizing with a bottleneck in this paper we present a heuristic method based on lagrangian relaxation for multilevel lot sizing when there is a single bottleneck facility a series of lagrangian relaxations one for each item in the product structure is imbedded in a branch and bound procedure the objective is to find a production schedule that fits within available capacity at minimum cost the method has two solution phases dual and primal in the dual phase of the procedure implied costs of setups and production are determined based on a tentative schedule the primal phase is repeated with these new prices and we iterate to reach a good solution the solution procedure is first tested on two special cases uncapacitated multilevel lot sizing and the capacitated single level multi item lot sizing problem the results show that the solution procedure can provide better solutions than some heuristics designed especially for those problems test results on the bottleneck problem indicate that good feasible solutions are found for problems too difficult to solve with exact methods

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

minimizing separable convex objectives on arbitrarily directed trees of variable upper bound constraints an extension of the economic order quantity eoq model to multi stage production distribution systems and the isotonic regression problem are known to be equivalent and to be solvable in on4 time the following specializations of the model are solvable in on log n time joint replenishment systems pure assembly systems nested policies in pure distribution systems nonnested policies in one warehouse multi retailer systems nested policies in multi item one warehouse multi retailer systems and systems in which the underlying network is a series parallel digraph we show here that a generalization of this problem is solvable in on log n time whenever the undirected version of the underlying network is a tree the algorithm we propose is used as a subroutine in an algorithm solving the eoq problem on general circuitless directed graphs the subject of a companion paper

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

approximation procedures for the one warehouse multi retailer system two heuristic procedures for a one warehouse multi retailer system are developed based on the accuracy desired the first heuristic evaluates a specified number of points the relative error is within a bound that approaches 1 2 ln 2 1 2 014 the complexity of the heuristic is on for a fixed number of evaluations although our bound only approaches the one of roundy 1985 when only a small number of points are evaluated our method is faster we show that the bound for our procedure and two bounds proposed by roundy 1985 are tight the second heuristic pertains to a class of polities called stationary interval policies for this class of policies we develop a fully polynomial time approximation scheme where the relative error is within e 0 and the computational effort increases as a linear function of 1 e computational experiments show that these heuristics perform well in practice

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

isotonic modeling with non differentiable loss functions with application to lasso regularization in this paper we present an algorithmic approach for fitting isotonic models under convex yet non differentiable loss functions it is a generalization of the greedy non regret approach proposed by luss and rosset 2014 for differentiable loss functions taking into account the sub gradiental extensions required we prove that our suggested algorithm solves the isotonic modeling problem while maintaining favorable computational and statistical properties as our suggested algorithm may be used for any non differentiable loss function we focus our interest on isotonic modeling for either regression or two class classification with appropriate log likelihood loss and lasso penalty on the fitted values this combination allows us to maintain the non parametric nature of isotonic modeling while controlling model complexity through regularization we demonstrate the efficiency and usefulness of this approach on both synthetic and real world data an implementation of our suggested solution is publicly available from the first author s website https sites google com site amichaipainsky software

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

principal gains and principal phases in the analysis of linear multivariable feedback systems the concepts of principal gain and principal phase are introduced for linear multivariable systems and their use in the analysis of feedback behavior is demonstrated a sufficient nyquist type stability criterion is presented in terms of these quantities and is used to characterize the robustness of the closed loop stability property when the system model is subjected to a linear perturbation either multiplicative or additive at any point in the feedback configuration the results presented are less conservative than those obtained via the small gain theorem

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

solutions of the cheeger problem via torsion functions abstract the cheeger problem for a bounded domain r n n 1 consists in minimizing the quotients e e among all smooth subdomains e and the cheeger constant h is the minimum of these quotients let p c 1 be the p torsion function that is the solution of torsional creep problem p p 1 in p 0 on where p u div u p 2 u is the p laplacian operator p 1 the paper emphasizes the connection between these problems we prove that lim p 1 p l 1 p h lim p 1 p l 1 1 p moreover we deduce the relation lim p 1 p l 1 c n lim p 1 p l where c n is a constant depending only of n and h explicitely given in the paper an eigenfunction u bv l of the dirichlet 1 laplacian is obtained as the strong l 1 limit as p 1 of a subsequence of the family p p l 1 p 1 almost all t level sets e t of u are cheeger sets and our estimates of u on the cheeger set e 0 yield b 1 h b 1 n e 0 h n where b 1 is the unit ball in r n for convex we obtain u e 0 1 e 0

fast algorithms for bipartite network flow we discuss network flow in a bipartite graph g with node sets s and t we show that karzanov s algorithm runs in time o s 2 t on g and that the mpm algorithm can be modified to run in the same bound for the common case that the degree of each node in t is bounded by a constant we show that a modified version of the mpm algorithm runs in time o s 3 s t on g the importance of these results comes from the common occurrence of problems that can be modeled and solved using network flow on bipartite graphs where s ll t in many applications involving bounded degree on the nodes in t t theta s 2 and our results reduce the best bounds on bipartite flow from o s 6 to o s 3 we use our results on bipartite flow to obtain the fastest known bounds on several combinatorial problems and mention several additional problems involving bipartite flow where s is much smaller than t

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 survey on the continuous nonlinear resource allocation problem our problem of interest consists of minimizing a separable convex and differentiable function over a convex set defined by bounds on the variables and an explicit constraint described by a separable convex function applications are abundant and vary from equilibrium problems in the engineering and economic sciences through resource allocation and balancing problems in manufacturing statistics military operations research and production and financial economics to subproblems in algorithms for a variety of more complex optimization models this paper surveys the history and applications of the problem as well as algorithmic approaches to its solution the most common techniques are based on finding the optimal value of the lagrange multiplier for the explicit constraint most often through the use of a type of line search procedure we analyze the most relevant references especially regarding their originality and numerical findings summarizing with remarks on possible extensions and future research 2006 elsevier b v all rights reserved

on the first exterior p harmonic steklov eigenvalue abstract in this short paper we study the sobolev function property of the rayleigh s quotient q inf u e 1 p u u p u q u p as a function of q 1 p for p p n 1 n p when p 1 n as well as the asymptotic behavior of positive solutions with minimal energy of the following problem p u 0 in u subject to u p 2 u u q 2 u on u to the first p harmonic steklov eigenpair on an exterior region u r n when n 3

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

uncertain multiple input multiple output systems with internal variable feedback there is given a linear time invariant multiple input multiple output mimo plant p with internal variables xi and outputs yi available for feedback due to uncertainty p is known only to be in a set p there are given tolerances on the overall system response functions tuv system output yu in response to command input ru to be satisfied pep it is shown how this mimo uncertainty problem may be replaced by a number of single input single output siso multiple loop uncertainty problems a suitable fixed point theorem guarantees that the solution of the siso problems solves the original mimo problem plant elements which appear singly in the mimo structure also appear singly in the siso structures a design example demonstrates how inner loop feedback may enormously decrease the effect of sensor noise at the mimo plant inputs extensions to disturbance attenuation under uncertainty and to uncertain time varying and non linear mimo plants are discussed

loss optimal monotone relabeling of noisy multi criteria data sets a method to relabel noisy multi criteria data sets is presented taking advantage of the transitivity of the non monotonicity relation to formulate the problem as an efficiently solvable maximum independent set problem a framework and an algorithm for general loss functions are presented and the flexibility of the approach is indicated by some examples showcasing the ease with which the method can handle application specific loss functions both didactical examples and real life applications are provided using the zero one the l1 and the squared loss functions as well as combinations thereof

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

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

improved rounding procedures for the discrete version of the capacitated eoq problem we consider a transportation problem where different products have to be shipped from an origin to a destination by means of vehicles with given capacity the production rate at the origin and the demand rate at the destination are constant over time and identical for each product the problem consists in deciding when to make the shipments and how to fill the vehicles with the objective of minimizing the sum of the average transportation and inventory costs at the origin and at the destination over an infinite horizon this problem is the well known capacitated eoq economic order quantity problem and has an optimal solution in closed form in this paper we study a discrete version of this problem in which shipments are performed only at multiples of a given minimum time it is known that rounding off the optimal solution of the capacitated eoq problem to the closest lower or upper integer value gives a tight worst case ratio of 2 while the best among the possible single frequency policies has a performance ratio of 5 3 we show that the 5 3 bound can be obtained by a single frequency policy based on a rounding procedure which considers classes of instances and for each class identifies a shipping frequency by rounding off in a different way the optimal solution of the capacitated eoq problem moreover we show that the bound can be reduced to 3 2 by using two shipping frequencies obtained by a rounding procedure in one class of instances only

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

robustness results in linear quadratic gaussian based multivariable control designs the robustness of control systems with respect to model uncertainty is considered using simple frequency domain criteria available and new results are derived under a common framework in which the minimum singular value of the return differences transfer matrix is the key quantity in particular robustness results associated with multivariable control systems designed on the basis of linear quadratic lq and the linear quadratic gaussian lqg design methodologies are presented

optimal and heuristic echelon r nq t policies in serial inventory systems with fixed costs this paper studies a periodic review serial inventory system in which echelon r nq t policies are implemented under such a policy each stage reviews its inventory in every t period and orders according to an echelon r nq policy two types of fixed costs are considered one is associated with each order batch q and the other is incurred for each inventory review the objective is to find the policy parameters such that the average total cost per period is minimized this paper provides a method for obtaining heuristic and optimal policy parameters the heuristic is based on minimizing lower and upper bounds on the total cost function these total cost bounds which are separable functions of the policy parameters are obtained in two steps first we decompose the total cost into costs associated with each stage which include a penalty cost for holding inadequate stock second we construct lower and upper bounds for the penalty cost by regulating downstream policy parameters to find the optimal solution we further construct cost bounds for each echelon a subsystem that includes a stage and all of its downstream stages by regulating holding and backorder cost parameters the echelon lower bound cost functions as well as the stage cost bounds generate bounds for the optimal solution in a numerical study we find that the heuristic is near optimal when the ratio of the fixed cost to the holding cost at the most downstream stage is large we also find that changing the optimal batch sizes may not affect the optimal reorder intervals or equivalently the delivery schedules under some conditions

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

active set algorithms for isotonic regression a unifying framework in this and subsequent papers we will show that several algorithms for the isotonic regression problem may be viewed as active set methods the active set approach provides a unifying framework for studying algorithms for isotonic regression simplifies the exposition of existing algorithms and leads to several new efficient algorithms we also investigate the computational complexity of several algorithms

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

multistage lot sizing problems via randomized rounding we study the classical multistage lot sizing problem that arises in distribution and inventory systems a celebrated result in this area is the 94 and 98 approximation guarantee provided by power of two policies in this paper we propose a simple randomized rounding algorithm to establish these performance bounds we use this new technique to extend several results for the capacitated lot sizing problems to the case with submodular ordering cost for the joint replenishment problem under a fixed base period model we construct a 95 8 approximation algorithm to the possibly dynamic optimal lot sizing policy the policies constructed are stationary but not necessarily of the power of two type this shows that for the fixed based planning model the class of stationary policies is within 95 8 of the optimum improving on the previously best known 94 approximation guarantee

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

coordination of production distribution networks with unbalanced leadtimes we investigate multiproduct multilocation production distribution networks with deterministic stationary demands most research on such systems assumes zero leadtimes methods using fixed reorder intervals that are power of two multiples of a base planning period have proved to be very successful for such systems the same methods apply to networks with positive leadtimes provided the leadtimes arebalanced in a specific sense we explore networks with general unbalanced leadtimes a key result is a lower bound on the cost of any feasible policy other results concern policy heuristics and their performance for a large class of networks we construct a policy whose cost is within 45 of the lower bound for general networks the performance guarantee is 1 02v 1 where is a number dependent on the network topology only in general the best performance bound is obtained by systematically reducing the order intervals derived from the corresponding zero leadtime system

heuristics for a one warehouse multiretailer distribution problem with performance bounds we investigate the one warehouse multiretailer distribution problem with traveling salesman tour vehicle routing costs we model the system in the framework of the more general production distribution system with arbitrary non negative monotone joint order costs we develop polynomial time heuristics whose policy costs are provably close to the cost of an optimal policy in particular we show that given a submodular function which is close to the true order cost then we can find a power of two policy whose cost is only moderately greater than the cost of an optimal policy since such submodular approximations exist for traveling salesman tour vehicle routing costs we present a detailed description of heuristics for the one warehouse multiretailer distribution problem we formulate a nonpolynomial dynamic program that computes optimal power of two policies for the one warehouse multiretailer system assuming only that the order costs are non negative monotone finally we perform computational tests which compare our heuristics to optimal power of two policies for problems of up to sixteen retailers we also perform computational tests on larger problems these tests give us insight into what policies one should employ

a relationship between sensitivity and stability of multivariable feedback systems both comparison sensitivity and robust stability analysis attempt to extend concepts which are well understood in scalar feedback systems to multivariable control system design although the two bodies of work have developed about a decade apart the basic philosophies are similar both attempt to maintain a system property sensitivity reduction versus stability in the presence of plant modeling errors this paper demonstrates how the similarity in philosophy translates into a similarity in results and interpretations

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

exact and heuristic solutions for a shipment problem with given frequencies we consider the problem of shipping several products from an origin to a destination when a discrete set of shipping frequencies is available in such a way that the sum of the transportation and inventory costs is minimized this problem which is known to be np hard has applications in transportation planning and in location analysis in this paper we derive dominance rules for the problem solutions that allow a tightening of the bounds on the problem variables and improve the efficiency of a known branch and bound algorithm moreover we present some heuristics and compare them with two different modifications of an eoq type algorithm for the solution of the problem with continuous frequencies

determining optimal reorder intervals in capacitated production distribution systems the problem of determining consistent and realistic reorder intervals in complex production distribution environments was formulated as a large scale nonlinear integer programming problem by maxwell and muckstadt maxwell w l j a muckstadt 1985 establishing consistent and realistic reorder intervals in production distribution systems oper res 33 6 november december 1316 1341 they show how the special structure of the problem permits its solution by a standard network flow algorithm in this paper we review the maxwell muckstadt model provide necessary and sufficient conditions that characterize the solution and show that the optimal partition of nodes in the production distribution network is invariant to an arbitrary scaling of the set up and holding cost parameters we consider two capacitated versions of the model one with a single constrained work center and the other with multiple constrained work centers for single constraint problems the invariance corollary provides a simp

feedback properties of multivariable systems the role and use of the return difference matrix for linear time invariant multivariable feedback systems the feedback properties of plant disturbance attenuation sensor noise response stability margins and sensitivity to plant and sensor variation are quantitatively related to the bode magnitude versus frequency plots of the singular values of the return difference matrix i l and of the associated inverse return difference matrix i l 1 implied fundamental limits of feedback performance are quantitatively described and design tradeoffs are discussed the penalty function in the stochastic linear quadratic gaussian lqg optimal control problem is found to be a weighted sum of the singular values with the weights determined by the quadratic cost and noise intensity matrices this enables systematic tuning of lqg cost and noise matrices so that the resulting optimal return difference and inversereturn difference meet inequality constraints derived from design specifications on feedback properties the theory has been used to synthesize a multivariable automatic controller for the longitudinal dynamics of an advanced fighter aircraft

the geometric maximum traveling salesman problem we consider the traveling salesman problem when the cities are points in n x211d d for some fixed d and distances are computed according to geometric distances determined by some norm we show that for any polyhedral norm the problem of finding a tour of maximum length can be solved in polynomial time if arithmetic operations are assumed to take unit time our algorithms run in time o nf 2 log n where f is the number of facets of the polyhedron determining the polyhedral norm thus for example we have o n2 log n algorithms for the cases of points in the plane under the rectilinear and sup norms this is in contrast to the fact that finding a minimum length tour in each case is np hard our approach can be extended to the more general case of quasi norms with a not necessarily symmetric unit ball where we get a complexity of o n2f 2 log n for the special case of two dimensional metrics with f 4 which includes the rectilinear and sup norms we present a simple algorithm with o n running time the algorithm does not use any indirect addressing so its running time remains valid even in comparison based models in which sorting requires n log n time the basic mechanism of the algorithm provides some intuition on why polyhedral norms allow fast algorithms complementing the results on simplicity for polyhedral norms we prove that for the case of euclidean distances in n x211d d for d 3 the maximum tsp is np hard this sheds new light on the well studied difficulties of euclidean distances

a new approach to the maximum flow problem all previously known efficient maximum flow algorithms work by finding augmenting paths either one path at a time as in the original ford and fulkerson algorithm or all shortest length augmenting paths at once using the layered network approach of dinic an alternative method based on the preflow concept of karzanov is introduced a preflow is like a flow except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out the method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths the algorithm and its analysis are simple and intuitive yet the algorithm runs as fast as any other known method on dense graphs achieving an o n 3 time bound on an n vertex graph by incorporating the dynamic tree data structure of sleator and tarjan we obtain a version of the algorithm running in o nm log n 2 m time on an n vertex m edge graph this is as fast as any known method for any graph density and faster on graphs of moderate density the algorithm also admits efficient distributed and parallel implementations a parallel implementation running in o n 2 log n time using n processors and o m space is obtained this time bound matches that of the shiloach vishkin algorithm which also uses n processors but requires o n 2 space

a faster algorithm solving a generalization of isotonic median regression and a class of fused lasso problems many applications in the areas of production signal processing economics bioinformatics and statistical learning involve a given partial order on certain parameters and a set of noisy observations of the parameters the goal is to derive estimated values of the parameters that satisfy the partial order that minimize the loss of deviations from the given observed values a prominent application of this setup is the well known isotonic regression problem where the partial order is a total order a commonly studied isotonic regression problem is the isotonic median regression imr where the loss function is a sum of absolute value functions on the deviations of the estimated values from the observation values we present here the most efficient algorithm to date for a generalization of imr with any convex piecewise linear loss function as well as for variant models that include ell 1 norm penalty separation regularization functions in the objective on the violation of the total order constraints suc

a parametric maximum flow algorithm for bipartite graphs with applications abstract suppose we are given a capacitated bipartite network g with node sets s and t in network g the arc capacities are not fixed values but are functions of a single parameter where is a continuous real variable our problem is to determine the minimum value of such that the maximum flow value in the corresponding network equals a given threshold for this problem an algorithm of time complexity o nm 2 log m n is presented where n is the number of nodes in s m is the number of nodes in t and n m examples are then given to show how to use this parametric algorithm to solve practical problems

the economic lot scheduling problem under power of two policy we present further analysis on the economic lot scheduling problem elsp without capacity constraints under power of two pot policy we explore its optimality structure and discover that the optimal objective value is piece wise convex by making use of the junction points of this function we derive an effective polynomial time search algorithm to secure a global optimal solution the conclusions of this research lay the foundation for deriving an efficient heuristic and also creates a benchmark for evaluating the quality of the heuristics for the conventional elsp under pot policy

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

reducing separable convex programs with tree constraints this paper describes a class of separable convex programs with tree constraints that has applications in production planning quality improvement and other related areas a reduction procedure is presented for solving this class of separable convex programs with n variables this reduction procedure determines an optimal solution to the convex problem by solving at most 2n simple convex subproblems with one variable hence this reduction procedure is an efficient approach for solving large scale convex programs of this sort

a fast parametric maximum flow algorithm and applications the classical maximum flow problem sometimes occurs in settings in which the arc capacities are not fixed but are functions of a single parameter and the goal is to find the value of the parameter such that the corresponding maximum flow or minimum cut satisfies some side condition finding the desired parameter value requires solving a sequence of related maximum flow problems in this paper it is shown that the recent maximum flow algorithm of goldberg and tarjan can be extended to solve an important class of such parametric maximum flow problems at the cost of only a constant factor in its worst case time bound faster algorithms for a variety of combinatorial optimization problems follow from the result