the lagrangian relaxation method for solving integer programming problems this article originally appeared in management science january 1981 volume 27 number 1 pp 1 18 published by the institute of management sciences one of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints dualizing the side constraints produces a lagrangian problem that is easy to solve and whose optimal value is a lower bound for minimization problems on the optimal value of the original problem the lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm this approach has led to dramatically improved algorithms for a number of important problems in the areas of routing location scheduling assignment and set covering this paper is a review of lagrangian relaxation based on what has been learned in the last decade

on continuous solutions of a class of systems of nonlinear functional difference equations with deviating argument abstract bounded continuous solutions on r of a class of systems of nonlinear functional difference equations with a complicated nonlinear deviation of the argument that depends on unknown functions are studied here

capacitated facility location separation algorithms and computational experience we consider the polyhedral approach to solving the capacitated facility location problem the valid inequalities considered are the knapsack cover flow cover effective capacity single depot and combinatorial inequalities the flow cover effective capacity and single depot inequalities form subfamilies of the general family of submodular inequalities the separation problem based on the family of submodular inequalities is np hard in general for the well known subclass of flow cover inequalities however we show that if the client set is fixed and if all capacities are equal then the separation problem can be solved in polynomial time for the flow cover inequalities based on an arbitrary client set and general capacities and for the effective capacity and single depot inequalities we develop separation heuristics an important part of these heuristics is based on the result that two specific conditions are necessary for the effective cover inequalities to be facet defining the way these results are stated indicates precisely how structures that violate the two conditions can be modified to produce stronger inequalities the family of combinatorial inequalities was originally developed for the uncapacitated facility location problem but is also valid for the capacitated problem no computational experience using the combinatorial inequalities has been reported so far here we suggest how partial output from the heuristic identifying violated submodular inequalities can be used as input to a heuristic identifying violated combinatorial inequalities we report on computational results from solving 60 medium size problems

a comparison of heuristics and relaxations for the capacitated plant location problem abstract approaches proposed in the literature for the capacitated plant location problem are compared the comparison is based on new theoretical and computational results the main emphasis is on relaxations in particular dominance relations among the various relaxations found in the literature are identified in the computational study the relaxations are compared as a function of various characteristics of the test problems several of these relaxations can be used to generate heuristic feasible solutions that are better than the classical greedy or interchange heuristics both in computing time and in the quality of the solutions found

a cross decomposition algorithm for capacitated facility location we present an implementation of the cross decomposition method to solve the capacitated facility location problem the method unifies benders decomposition and lagrangean relaxation into a single framework that involves successive solutions to a transportation problem and a simple plant location problem computational tests on problems with as many as 100 candidate facility locations and 200 customers provide evidence that the approach is superior to several other methods the specialized cross decomposition program is on a common set of test problems about ten times faster than several recently developed programs we also present a simple algorithm to strengthen benders cuts

on the recursive sequence xn 1 α xn 1xn in this paper we study the global stability the permanence and the oscillation character of the recursive sequence xn 1 xn 1xn n 0 1 where is a negative number and the initial conditions x 1 and x0 are negative numbers

a lagrangian heuristic for the capacitated plant location problem with single source constraints abstract in this paper we look at the lagrangian relaxations of the capacitated plant location problem with single source constraints in terms of the bounds and solution techniques we propose a heuristic based on a lagrangian relaxation of the problem we also show how this heuristic can be used to solve other combinatorial problems that are related to this problem such as the concentrator location problem and the bin packing problem

branch and price column generation for solving huge integer programs we discuss formulations of integer programs with a huge number of variables and their solution by column generation methods i e implicit pricing of nonbasic variables to generate new columns or to prove lp optimality at a node of the branch and bound tree we present classes of models for which this approach decomposes the problem provides tighter lp relaxations and eliminates symmetry we then discuss computational issues and implementation of column generation branch and bound algorithms including special branching rules and efficient ways to solve the lp relaxation we also discuss the relationship with lagrangian duality

extension of the blahut arimoto algorithm for maximizing directed information in this paper we extend the blahut arimoto algorithm for maximizing massey s directed information the algorithm can be used for estimating the capacity of channels with delayed feedback where the feedback is a deterministic function of the output in order to maximize the directed information we apply the ideas from the regular blahut arimoto algorithm i e the alternating maximization procedure to our new problem we provide both upper and lower bound sequences that converge to the optimum global value our main insight in this paper is that in order to find the maximum of the directed information over a causal conditioning probability mass function one can use a backward index time maximization combined with the alternating maximization procedure we give a detailed description of the algorithm showing its complexity and the memory needed and present several numerical examples

capacity of a post channel with and without feedback we consider finite state channels where the state of the channel is its previous output we refer to these as previous output is the state post channels we first focus on post channels these channels have binary inputs and outputs where the state determines if the channel behaves as a z or an s channel both with parameter we show that the nonfeedback capacity of the post channel equals its feedback capacity despite the memory of the channel the proof of this surprising result is based on showing that the induced output distribution when maximizing the directed information in the presence of feedback can also be achieved by an input distribution that does not utilize the feedback we show that this is a sufficient condition for the feedback capacity to equal the nonfeedback capacity for any finite state channel we show that the result carries over from the post channel to a binary post channel where the previous output determines whether the current channel will be binary with parameters a b or b a finally we show that in general feedback may increase the capacity of a post channel

an algorithm for solving large capacitated warehouse location problems abstract in this paper we present a lower bound for the capacitated warehouse location problem based upon lagrangean relaxation of a mixed integer formulation of the problem feasible solution exclusion constraints are used together with problem reduction tests derived from both the original problem and the lagrangean relaxation by incorporating the lower bound and the reduction tests into a tree search procedure we are able to solve problems involving up to 500 potential warehouse locations and 1000 customers

using central prices in the decomposition of linear programs abstract this paper deals with a decomposition technique for linear programs which proposes a new treatment of the master program in the classical dantzig wolfe algorithm this approach exploits a the relationship between the master program and the minimization of a convex piecewise linear function and b a recently proposed cutting plane algorithm for convex nondifferentiable optimization the new algorithm seeks to achieve deep cuts by generating them at or near to the analytic centre of a set of localization containing the solution therefore each iteration entails a sizeable decrease of the set of localization and the overall algorithm shows fast convergence this algorithm has been used to solve reputedly difficult convex nondifferentiable optimization problems its implementation as a decomposition algorithm for large scale structured linear program seems to be a promising alternative to the standard dantzig wolfe approach which suffers from very slow convergence in many practical problems the new approach differs from the dantzig wolfe algorithm through the fact that it does not use prices corresponding to an optimal combination of the past proposals of the subproblem but rather central prices i e analytic centres which balance them all this property seems to explain the good behaviour of the new algorithm

a heuristic lagrangean algorithm for the capacitated plant location problem abstract lagrangean techniques have been widely applied to the uncapacitated plant location problem and in some cases they have proven to be successfull even when capacitated problems with additional constraints are taken into account in our paper we study the application of these techniques to the capacitated plant location problem when the model considered is a pure integer one several lagrangean decompositions are considered and for some of them heuristic algorithms have been designed to solve the resulting lagrangean subproblems the heuristics consisting of a two phase procedure the first location phase defines a set of multipliers from the analysis of the dual lp relaxation and makes a choice of the plants considering the resulting subproblems as a particular case of the general assignment problems several heuristics have been studied for this second phase based either on a decomposition of knapsack type subproblems through a definition of a set of penalties or of looking into the duality gap and trying to reduce it computational experience is reported

benders decomposition without separability a computational study for capacitated facility location problems abstract benders is one of the most famous decomposition tools for mathematical programming and it is the method of choice e g in mixed integer stochastic programming its hallmark is the capability of decomposing certain types of models into smaller subproblems each of which can be solved individually to produce local information notably cutting planes to be exploited by a centralized master problem as its name suggests the power of the technique comes essentially from the decomposition effect i e the separability of the problem into a master problem and several smaller subproblems in this paper we address the question of whether the benders approach can be useful even without separability of the subproblem i e when its application yields a single subproblem of the same size as the original problem in particular we focus on the capacitated facility location problem in two variants the classical linear case and a congested case where the objective function contains convex but non separable quadratic terms we show how to embed the benders approach within a modern branch and cut mixed integer programming solver addressing explicitly all the ingredients that are instrumental for its success in particular we discuss some computational aspects that are related to the negative effects derived from the lack of separability extensive computational results on various classes of instances from the literature are reported with a comparison with the state of the art exact and heuristic algorithms the outcome is that a clever but simple implementation of the benders approach can be very effective even without separability as its performance is comparable and sometimes even better than that of the most effective and sophisticated algorithms proposed in the previous literature

capacity of the trapdoor channel with feedback we establish that the feedback capacity of the trapdoor channel is the logarithm of the golden ratio and provide a simple communication scheme that achieves capacity as part of the analysis we formulate a class of dynamic programs that characterize capacities of unifilar finite state channels the trapdoor channel is an instance that admits a simple closed form solution

on the difference equation xn 1 α xn 1xn in this paper we study the behavior of the solutions of the difference equation x n 1 a x n 1x n n 0 1 where a is a negative number included are results which considerably improve and correct those in the recently published paper a e hamza on the recursive sequence x n 1 a x n 1x n j math anal appl 322 2006 668 674 we also refute conjecture 2 in g ladas open problems and conjectures j difference equ appl 7 2 2001 477 482

on a nonlinear second order system of difference equations we prove that all positive solutions of the next system of difference equationsx n 1 a y n px n 1 q y n 1 a x n py n 1 q email protected n 0 where parameters a p and q are positive numbers are bounded if one of the following conditions is satisfied p 2 4 or p 1 q q 1 completely describing the boundedness character of the system in this case

asymptotic and oscillatory properties of the fourth order nonlinear difference equations in this article a class of fourth order difference equations with quasi differences and a deviating argument is considered we state new oscillation theorems for the super linear and for the half linear case and we complete the existing results in the literature

valid inequalities and facets of the capacitated plant location problem recently several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods and polyhedral characterizations of integer programming problems in this paper we investigate the polyhedral structure of the capacitated plant location problem our purpose is to identify facets and valid inequalities for a wide range of capacitated fixed charge problems that contain this prototype problem as a substructure

extensions to a lagrangean relaxation approach for the capacitated warehouse location problem abstract in this paper we present a lower bound for the capacitated warehouse location problem based upon the lagrangean relaxation of a mixed integer formulation of the problem where we use subgradient optimisation in an attempt to maximise this lower bound problem reduction tests based upon this lower bound and the original problem are given incorporating this bound and the reduction tests into a tree search procedure enables us to solve problems involving up to 50 warehouses and 150 customers

application of difference equations in divisibility abstract we present an easy applicable method which for a large class of integer sequences a n shows when all their terms are divisible by a natural number m 1

global stability of some symmetric difference equations suppose r 0 1 m n and 1 k1 k2 k2m 1 and let s2m 1 1 2 2m 1 we show that every positive solution to the difference equation yn p2m 12m 1 yn k1r yn k2r yn k2m 1r p2m2m 1 yn k1r yn k2r yn k2m 1r n n0 where p2m 12m 1 x1 x2 x2m 1 r 1rodd2m 1 t1 t2 tr s2m 1t1 t2 trxt1xt2 xtr and p2m2m 1 x1 x2 x2m 1 1 r 2reven2m t1 t2 tr s2m 1t1 t2 trxt1xt2 xtr converges to one this result confirms a quite recent conjecture posed by liu and yang 2010 in 10 we also prove another result regarding a related equation

lot size models with back logging strong reformulations and cutting planes we examine mixed integer programming reformulations of the uncapacitated lot sizing problem with backlogging first we consider the effect of using a standard reformulation technique for fixed charge network flow problems which involves the introduction of new variables leading to a known plant location reformulation and a shortest path reformulation each of these reformulations is strong in the sense that its linear programming relaxation solves the lot sizing problem secondly we attempt to treat the problem in the space of the original variables we give an implicit description of the convex hull of solutions and show how the problem of finding a violated cutting plane can be solved as a linear program we also describe a family of strong valid inequalities which can be generated rapidly by a heuristic and which have proved effective in a cut generation algorithm the efficiency of both the shortest path formulation and the cutting plane algorithm have been tested on a series of multi item capacitated lot sizing problems with backlogging near optimal solutions have been found to problems with 8 periods and up to 100 times

upper and lower bounds for the single source capacitated location problem abstract the single source capacitated location problem is considered given a set of potential locations and the plant capacities it must be decided where and how many plants must be open and which clients must be assigned to each open plant a lagrangean relaxation is used to obtain lower bounds for this problem upper bounds are given by lagrangean heuristics followed by search methods and by one tabu search metaheuristic computational experiments on different sets of problems are presented

lagrangian heuristics for the two echelon single source capacitated facility location problem facility location problems form an important class of integer programming problems with application in the distribution and transportation industries in this paper we are concerned with a particular type of facility location problem in which there exist two echelons of facilities each facility in the second echelon has limited capacity and can be supplied by only one facility or depot in the first echelon each customer is serviced by only one facility in the second echelon the number and location of facilities in both echelons together with the allocation of customers to the second echelon facilities are to be determined simultaneously we propose a mathematical model for this problem and consider six heuristics based on lagrangian relaxation for its solution to solve the dual problem we make use of a subgradient optimization procedure we present numerical results for a large suite of test problems these indicate that the lower bounds obtained from some relaxations have a duality gap which frequently is one third of the one obtained from traditional linear programming relaxation furthermore the overall solution time for the heuristics are less than the time to solve the lp relaxation

an exact method for the two echelon single source capacitated facility location problem facility location problems form an important class of integer programming problems with applications in the telecommunication distribution and transportation industries in this paper we are conc

global stability and asymptotics of some classes of rational difference equations in this note we prove that the positive solutions of some classes of rational difference equations are globally asymptotically stable using a berg s result we also find asymptotics of some solutions of these equations

global stability of a difference equation with maximum we prove that every positive solution to the difference equationx n maxa 1x n 1 a 1 a 2x n 2 a 2 a kx n k a k email protected n 0 where email protected n a i 0 a email protected 0 1 i 1 k converges to the following quantity maxa 1 1 a 1 1 a k 1 a k 1 confirming a quite recent conjecture of interest we also prove another result on global convergence which concerns some cases when not all a i i 1 k belong to the interval 0 1

decomposition and nondifferentiable optimization with the projective algorithm this paper deals with an application of a variant of karmarkar s projective algorithm for linear programming to the solution of a generic nondifferentiable minimization problem this problem is closely related to the dantzig wolfe decomposition technique used in large scale convex programming the proposed method is based on a column generation technique defining a sequence of primal linear programming maximization problems associated with each problem one defines a weighted potential function which is minimized using a variant of the projective algorithm when a point close to the minimum of the potential function is reached a corresponding point in the dual space is constructed which is close to the analytic center of a polytope containing the solution set of the nondifferentiable optimization problem an admissible cut of the polytope corresponding to a new supporting hyperplane of the epigraph of the function to minimize is then generated at this approximate analytic center in the primal space this new cut translates into a new column for the associated linear programming problem the algorithm has performed well on a set of convex nondifferentiable programming problems

capacity and zero error capacity of ising channels simple binary symmetric one and two dimensional channel systems called ising channels are investigated the one dimensional version is a certain binary finite state channel with input intersymbol interference its zero error capacity is determined together with lower and upper bounds to its shannon capacity for the two dimensional channel system a lower bound is obtained for the zero error capacity and an efficient zero error coding scheme is developed

lagrangean heuristics for location problems abstract in this paper we present a framework for developing lagrangean heuristics heuristics based upon lagrangean relaxation and subgradient optimisation with respect to location problems computational results are given for four different location problems p median uncapacitated warehouse location capacitated warehouse location and capacitated warehouse location with single source constraints these results indicate that the framework presented in this paper is robust i e it gives good quality solutions for each of these different location problems

a multi exchange heuristic for the single source capacitated facility location problem we present a very large scale neighborhood vlsn search algorithm for the capacitated facility location problem with single source constraints the neighborhood structures are induced by customer multi exchanges and by facility moves we consider both traditional single customer multi exchanges detected on a suitably definedcustomer improvement graph and more innovative multicustomer multi exchanges detected on afacility improvement graph dynamically built through the use of a greedy scheme computational results for some benchmark instances are reported that demonstrate the effectiveness of the approach for solving large scale problems a further test on real data involving an italian factory is also presented

boundedness character of a class of difference equations abstract a complete picture regarding the boundedness character of positive solutions to the following difference equation x n max a x n 1 p x n k p n n 0 where k 2 and the parameters a and p are positive real numbers is given in particular for the case p k 1 0 k k k 1 k 1 we prove that all solutions to the equation are bounded we also present corresponding results concerning the following closely related difference equation x n a x n 1 p x n k p n n 0

local existence of lipschitz continuous solutions of systems of nonlinear functional equations with iterated deviations abstract some conditions which guarantee the local existence of lipschitz continuous solutions of some systems of nonlinear functional equations with complicated deviations dependent of unknown functions are presented

a repeated matching heuristic for the single source capacitated facility location problem facility location models form an important class of integer programming problems with application in many areas such as the distribution and transportation industries an important class of solution methods for these problems are so called lagrangean heuristics which have been shown to produce high quality solutions and which are at the same time robust the general facility location problem can be divided into a number of special problems depending on the properties assumed in the capacitated location problem each facility has a specific capacity on the service it provides we describe a new solution approach for the capacitated facility location problem when each customer is served by a single facility the approach is based on a repeated matching algorithm which essentially solves a series of matching problems until certain convergence criteria are satisfied the method generates feasible solutions in each iteration in contrast to lagrangean heuristics where problem dependent heuristics must be used to construct a feasible solution numerical results show that the approach produces solutions which are of similar and often better than those produced using the best lagrangean heuristics

a branch and price algorithm for the capacitated facility location problem the capacitated facility location problem cflp is a well known combinatorial optimization problem with applications in distribution and production planning it consists in selecting plant sites from a finite set of potential sites and in allocating customer demands in such a way as to minimize operating and transportation costs a number of solution approaches based on lagrangean relaxation and subgradient optimization has been proposed for this problem subgradient optimization does not provide a primal fractional optimal solution to the corresponding master problem however in order to compute optimal solutions to large or difficult problem instances by means of a branch and bound procedure information about such a primal fractional solution can be advantageous in this paper a stabilized column generation method is therefore employed in order to solve a corresponding master problem exactly the column generation procedure is then employed within a branch and price algorithm for computing optimal solutions to the cflp computational results are reported for a set of larger and difficult problem instances

an efficient tabu search procedure for the p median problem abstract in this paper we present a new solution heuristic for the p median problem the algorithm is based on tabu search principles and uses short term and long term memory as well as strategic oscillation and random tabu list sizes our proposed procedure is compared with two other move heuristics a well known interchange heuristic and a recent hybrid heuristic in computational tests on networks ranging in size up to 500 nodes the new heuristic is found to be superior with respect to the quality of solutions produced

on the system xn 1 ynx n k yn k 1 an bnynxn k yn 1 xnyn k xn k 1 cn dnxnyn k it is shown that the system of difference equationsx n 1 y nx n ky n k 1 a n b ny nx n k y n 1 x ny n kx n k 1 c n d nx ny n k email protected n 0 where email protected n the sequences a n b n c n d n email protected n 0 and the initial values x i y i i 0 email protected are real numbers can be solved in closed form asymptotic behavior of well defined solutions of the system for the case when all the sequences a n b n c n and d n are constant is studied in detail

tighter bounds on the capacity of finite state channels via markov set chains the theory of markov set chains is applied to derive upper and lower bounds on the capacity of finite state channels that are tighter than the classic bounds by gallager the new bounds coincide and yield single letter capacity characterizations for a class of channels with the state process known at the receiver including channels whose long term marginal state distribution is independent of the input process analogous results are established for finite state multiple access channels

finite state channels with time invariant deterministic feedback we consider capacity of discrete time channels with feedback for the general case where the feedback is a time invariant deterministic function of the output samples under the assumption that the channel states take values in a finite alphabet we find a sequence of achievable rates and a sequence of upper bounds on the capacity the achievable rates and the upper bounds are computable for any n and the limits of the sequences exist we show that when the probability of the initial state is positive for all the channel states then the capacity is the limit of the achievable rate sequence we further show that when the channel is stationary indecomposable and has no intersymbol interference isi its capacity is given by the limit of the maximum of the normalized directed information between the input xn and the output yn i e c limnrarrinfin 1 n max i xnrarryn where the maximization is taken over the causal conditioning probability q xnparzn 1 defined in this paper the main idea for obtaining the results is to add causality into gallager s results on finite state channels the capacity results are used to show that the source channel separation theorem holds for time invariant determinist feedback and if the state of the channel is known both at the encoder and the decoder then feedback does not increase capacity

an exact algorithm for the capacitated facility location problems with single sourcing facility location problems are often encountered in many areas such as distribution transportation and telecommunication we describe a new solution approach for the capacitated facility location problem in which each customer is served by a single facility an important class of heuristic solution methods for these problems are lagrangian heuristics which have been shown to produce high quality solutions and at the same time be quite robust a primal heuristic based on a repeated matching algorithm which essentially solves a series of matching problems until certain convergence criteria are satisfied is incorporated into the lagrangian heuristic finally a branch and bound method based on the lagrangian heuristic is developed and compared computationally to the commercial code cplex the computational results indicate that the proposed method is very efficient

on the system of difference equations xn cnyn 3 an bnyn 1xn 2y n 3 yn γnxn 3 αn βnxn 1yn 2 xn 3 we show that the system of difference equationsx n c ny n 3a n b ny n 1x n 2y n 3 y email protected nx n email protected email protected nx n 1y n 2x n 3 email protected n 0 where the sequences a n b n c n a n b n c n email protected n 0 and initial values x i y i email protected 1 2 3 are real numbers such that c n 0 c n 0 email protected n 0 can be solved in closed form and for the case when all the sequences a n b n c n a n b n and c n are constant we describe the asymptotic behavior of well defined solutions of the system

on positive solutions of some classes of max type systems of difference equations global attractivity results on positive solutions of some special cases of the next max type system of difference equations xn i max1 xn l1 i i p1 i j1 1m1 i xn j1 i1 sj1 i1 qj1 i1 xn lri i i pri i jri 1mri i xn jri iri sjri iri qjri iri n n0 where ri n i 1 k k n sjt it 1 k qjt it 0 i 1 k t 1 ri and pj i 0 1 mj i n i 1 k j 1 ri are presented we also present some results on the behavior of positive solutions of the difference equation xn max p1xn 1 pkxn k xn max p1xn 1 pkxn k where pj j 1 k are nonnegative numbers

behaviour of solutions of some linear functional equations at infinity relationship between the boundedness and the existence of a finite limit at the infinity of solutions of the linear functional equationx f t email protected t f t where email protected c f r r is a continuous increasing function satisfying the condition f t t email protected r and f is a continuous function such that there is a finite limit lim t f t is studied here by using obtained results we study behaviour of bounded solutions of the functional equationx f k t email protected email protected ix f i t f t where a i i 1 email protected are real numbers such that i 0 k email protected i 1 functions f and f satisfy above mentioned conditions and lim t f t 0

on a class of third order nonlinear difference equations this paper studies the boundedness character of the positive solutions of the difference equation x n 1 a x n px n 1 qx n 2 r n n 0 where the parameters a p q and r are positive numbers

inverse optimization an application to the capacitated plant location problem lagrangean techniques have had wide application to the optimization of discrete optimization problems inverse optimization refers to the fact that each time a lagrangean calculation is made for a specific problem with a given resources vector an optimal solution is obtained for a related problem with a suitably adjusted resources vector this property is studied in depth for the capacitated plant location problem and new parametric methods for that problem are suggested computational experience is reported

difference equations and nonexistence of solutions of an integral equation abstract by using difference equations we give some easily applicable sufficient conditions for the non existence of nonnegative solutions satisfying some additional conditions of the integral equation f x x a x t b x t f t p d t where p 1 and x is a measurable space with probability measure

on asymptotic behaviour of the difference equation xn 1 α xn kxn in this paper we investigate the global stability and the periodic character of positive solutions of the difference equationx n email protected x n kx nunder specified conditions

a randomized algorithm for the capacity of finite state channels inspired by ideas from the field of stochastic approximation we propose a randomized algorithm to compute the capacity of a finite state channel with a markovian input when the mutual information rate of the channel is concave with respect to the chosen parameterization the proposed algorithm proves to be convergent to the capacity of the channel almost surely with the derived convergence rate we also discuss the convergence behavior of the algorithm without the concavity assumption

the uniform distribution as a universal prior in this correspondence we discuss the properties of the uniform prior as a universal prior i e a prior that induces a mutual information that is simultaneously close to the capacity for all channels we determine bounds on the amount of the mutual information loss in using the uniform prior instead of the capacity achieving prior specifically for the class of binary input channels with any output alphabet we show that the z channel has the minimal mutual information with uniform prior out of all channels with a given capacity from this we conclude that the degradation of the mutual information with respect to the capacity is at most 0 011 bit and as was shown previously at most 6 a related result is that the capacity achieving prior for any channel is not far from uniform some of these results are extended to channels with nonbinary input

the capacity of channels with feedback in this paper we introduce a general framework for treating channels with memory and feedback first we prove a general feedback channel coding theorem based on massey s concept of directed information second we present coding results for markov channels this requires determining appropriate sufficient statistics at the encoder and decoder we give a recursive characterization of these sufficient statistics third a dynamic programming framework for computing the capacity of markov channels is presented fourth it is shown that the average cost optimality equation acoe can be viewed as an implicit single letter characterization of the capacity fifth scenarios with simple sufficient statistics are described sixth error exponents for channels with feedback are presented

dynamic programming and strong bounds for the 0 1 knapsack problem two new algorithms recently proved to outperform all previous methods for the exact solution of the 0 1 knapsack problem this paper presents a combination of such approaches where in additi on valid inequalities are generated and surrogate relaxed and a new initial core problem is adopted the algorithm is able to solve all classical test instances with up to 10 000 variables in less than 0 2 seconds on a hp9000 735 99 computer the c language implementation of the algorithm is available on the internet

a hypergraph multi exchange heuristic for the single source capacitated facility location problem abstract in this paper we introduce a large scale neighborhood search procedure for solving the single source capacitated facility location problem sscflp the neighborhood structures are induced by innovative split multi customer multi exchanges where clusters of customers assigned to one facility can be moved simultaneously to multiple destination facilities and vice versa to represent these exchanges we use two types of improvement hypergraphs the improvement hypergraphs are built dynamically and the moving customers associated with each hyperedge are selected by solving heuristically a suitably defined mixed integer program we develop a hypergraph search framework including forward and backward procedures to identify improving solutions efficiently our proposed algorithm can obtain improving moves more quickly and even find better solutions than a traditional multi exchange heuristic ahuja orlin pallottino scuttella 2004 in addition when compared with the kernel search algorithm guastaroba speranza 2014 which at present is the most effective for solving sscflp our algorithm is not only competitive but can find better solutions or even the best known solution to some very large scale benchmark instances from the literature

global stability of a max type difference equation we show that every positive solution to the difference equationx n maxa 1x n p 1 a 1 a 2x n p 2 a 2 a kx n p k a k n n 0 where p i i 1 k are natural numbers such that 1 0 a i 1 1 i 1 k converges to max 1 i ka i 1 a i 1 this result improves and complements the main result in our recent note s stevic global stability of a difference equation with maximum appl math comput 210 2009 525 529 since it also considers the case when a i 1 0 i 1 k

on the recursive sequencexn 1 α xn 1 xn abstract we study the global stability the boundedness character and the periodic nature of the positive solutions of the difference equationxn 1 xn 1 xn where 0 and where the initial conditionsx 1andx0are arbitrary positive real numbers

a heuristic lagrangean algorithm for the capacitated plant location problem a comment abstract this is a comment on the paper a heuristic lagrangean relaxation algorithm for the capacitated plant location problem by j barcelo and j casanovas the authors have a single source version of the problem where they provide a necessary and sufficient condition for the feasibility of the problem we show that the condition as shown by the authors is only a necessary condition and not a sufficient condition using an example

a lagrangean relax and cut approach for the two stage capacitated facility location problem abstract the two stage capacitated facility location problem is to find the optimal locations of depots to serve customers with a given demand the optimal assignment of customers to depots and the optimal product flow from plants to depots in order to solve this problem a lagrangean heuristic is proposed which is based on the relaxation of the capacity constraints the resulting lagrangean subproblem is an uncapacitated facility location problem with an aggregate capacity constraint and can be solved efficiently by branch and bound methods the best bound available from this relaxation is computed by means of a weighted dantzig wolfe decomposition approach feasible solutions are constructed from those of the lagrangean subproblems by applying simple reassignment procedures furthermore the relaxation is strengthened by adding and dualizing valid inequalities which are violated by the fractional primal solution of the dual master program computational results and comparisons with some other bounds and heuristics for this problem are presented

the capacity of finite state markov channels with feedback we consider a class of finite state markov channels with feedback we first introduce a simplified equivalent channel model and then construct the optimal stationary and nonstationary input processes that maximize the long term directed mutual information furthermore we give a sufficient condition under which the channel s shannon capacity can be achieved by a stationary input process the corresponding converse coding theorem and direct coding theorem are proved

approximate solutions to large scale capacitated facility location problems this article presents a heuristic algorithm for solving large scale capacitated facility location problems the algorithm consists of solving a single source facility location problem and then solving a minimum cost network problem the single source problem is solved by applying an iterative procedure in which lower bounds on the optimal objective value are generated through lagrangian relaxation techniques upper bounds on the optimal objective value are obtained by modifying the lagrangian solution to restore feasibility the iterative procedure is terminated when the difference between the two bounds is within a prespecified tolerance further improvement of the objective value is attained by subjecting the single source solution to a network optimization code the heuristic algorithm has been tested on over 300 problems derived in one form or another from the kuehn hamburger test problems the testing has also been undertaken on the problems of van roy which were generated using netgen computational results indicate that our approach can be effective in producing near optimal solutions for problems with as many as 1000 customers and 100 potential facility locations most problems converged within 3 percent of the optimal value requiring only a modest amount of computation on an ibm 3090

efficient production distribution system design the production distribution system design is an integral part of the general supply chain design this paper proposes a novel solution methodology for this problem that is based on lagrangean relax

accelerating benders decomposition algorithmic enhancement and model selection criteria this paper proposes methodology for improving the performance of benders decomposition when applied to mixed integer programs it introduces a new technique for accelerating the convergence of the algorithm and theory for distinguishing good model formulations of a problem that has distinct but equivalent mixed integer programming representations the acceleration technique is based upon selecting judiciously from the alternate optima of the benders subproblem to generate strong or pareto optimal cuts this methodology also applies to a much broader class of optimization algorithms that includes dantzig wolfe decomposition for linear and nonlinear programs and related cutting plane type algorithms that arise in resource directive and price decomposition when specialized to network location problems this cut generation technique leads to very efficient algorithms that exploit the underlying structure of these models in discussing the proper formulation of mixed integer programs we suggest criteri

the boundedness character of two stevic type fourth order difference equations abstract the boundedness character of positive solutions of two nonlinear fourth order difference equations which are particular cases of two large classes of difference equations by stevic are studied in this paper

on the max type equation xn 1 max axn xn 2 abstract we show that every well defined solution of the difference equation x n 1 max a x n x n 2 n n 0 where a r is eventually periodic with period three

a coding theorem for a class of stationary channels with feedback a coding theorem is proved for a class of stationary channels with feedback in which the output yn f xn m n zn m n is the function of the current and past m symbols from the channel input xn and the stationary ergodic channel noise zn in particular it is shown that the feedback capacity is equal to limnrarr infin supp x n y n 1 1 n i xn rarr yn where i xn rarr yn sigmai 1 n i xi yi yi 1 denotes the massey directed information from the channel input to the output and the supremum is taken over all causally conditioned distributions p xn yn 1 pii 1 n p xi xi 1 yi 1 the main ideas of the proof are a classical application of the shannon strategy for coding with side information and a new elementary coding technique for the given channel model without feedback which is in a sense dual to gallager s lossy coding of stationary ergodic sources a similar approach gives a simple alternative proof of coding theorems for finite state channels by yang kavcic tatikonda chen berger and permuter weissman goldsmith

long term behavior of positive solutions of a system of max type difference equations abstract some sufficient conditions which guaranty the attractivity of all positive solutions of the next system of difference equations x n max a y n 1 p j 2 k x n j q j y n max a x n 1 p j 2 k y n j q j n n 0 where a p 0 and q j j 2 k are nonnegative numbers are given the boundedness character of positive solutions of the system is also studied

capacitated facility location valid inequalities and facets we examine the polyhedral structure of the convex hull of feasible solutions of the capacitated facility location problem in particular we derive necessary and sufficient conditions for a family of effective capacity inequalities to be facet defining and further results on a more general family called submodular inequalities