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

a distributed newton method for network utility maximization i algorithm most existing works use dual decomposition and first order methods to solve network utility maximization num problems in a distributed manner which suffer from slow rate of convergence properties this paper develops an alternative distributed newton type fast converging algorithm for solving num problems by using novel matrix splitting techniques both primal and dual updates for the newton step can be computed using iterative schemes in a decentralized manner we propose a stepsize rule and provide a distributed procedure to compute it in finitely many iterations the key feature of our direction and stepsize computation schemes is that both are implemented using the same distributed information exchange mechanism employed by first order methods we describe the details of the inexact algorithm here and in part ii of this paper we show that under some assumptions even when the newton direction and the stepsize in our method are computed within some error due to finite truncation of the iterative schemes the resulting objective function value still converges superlinearly in terms of primal iterations to an explicitly characterized error neighborhood simulation results demonstrate significant convergence rate improvement of our algorithm relative to the existing first order methods based on dual decomposition

algorithms for a multi level network optimization problem in this paper we work on a multi level network optimization problem that integrates into the same model important aspects of i discrete facility location ii topological network design and iii network dimensioning potential applications for the model are discussed stressing its growing importance the multi level network optimization problem treated is defined and a mathematical programming formulation is presented we make use of a branch and bound algorithm based on lagrangean relaxation lower bounds to introduce some new powerful auxiliary algorithms to exactly solve the problem we conduct a set of computational experiments that indicate the quality of the proposed approach

insuring causality of frequency response functions with hysteretic damping when calculating the temporal response h t of a system responding linearly to an external impulsive force applied at t 0 the availability of personal computers that calculate fourier transforms may make it convenient to first calculate the frequency response function h of the system and then obtain h t as the inverse fourier transform of h dissipation can be accounted for by multiplying the stiffness and elastic quantities in h by the complex factor 1 i assuming hysteretic damping with loss factor however this makes h a complex function of frequency whose inverse transform may be neither real nor causal as is well known the inverse transform can be made real by setting h h where indicates complex congugate or replacing with sgn where sgn 1 but this may not lead to causality it is suggested that causality be insured by expressing the temporal function simply as h t 1 sgn t h t where h t is the inverse fourier transform of either the real or the

projected newton methods and optimization of multicommodity flows a superlinearly convergent newton like method for linearly constrained optimization problems is adapted for solution of multicommodity network flow problems of the type arising in communication and transportation networks we show that the method can be implemented approximately by making use of conjugate gradient iterations without the need to compute explicitly the hessian matrix preliminary computational results suggest that this type of method is capable of yielding highly accurate solutions of nonlinear multicommodity flow problems far more efficiently than any of the methods available at present

generating alternative mixed integer programming models using variable redefinition dropping the complicating constraints in a mixed integer linear program often yields a special structure subproblem that can be reformulated using a different set of decision variables once the new variables have been identified the entire problem can be reformulated in terms of the new variables we develop a theory of variable redefinition based on relating the two sets of decision variables by a linear transformation and describe methods for reformulating the special structure problem the reformulated models have a more accurate linear relaxation than the problems from which they were derived an important property within the context of linear programming based branch and bound modeling approaches

dynamic dispersion curves for non homogeneous anisotropic beams with cross sections of arbitrary geometry abstract to accurately predict the dynamic behaviour of spanwise uniform non homogeneous anisotrophic beams with arbitrary cross sectional geometry is an important part of engineering analysis classical one dimensional beam theory fails in this endeavour and development of more sophisticated one dimensional theories is an important task a basis for development of improved one dimensional theory is the dispersion curve analysis a means to find the dispersion curves is discussed in this paper the code is developed to calculate the dispersion curves as well as corresponding mode shapes the code is based on finite element discretization over the cross sectional domain use of this code provides benchmark results for the testing of any existing one dimensional theory as well as facilitating the construction of new one dimensional theories dispersion information can also be useful in non destructive evaluation as an example of a case where non classical effects are particularly important results obtained from this code for an i beam are presented qualitative comparisons with the well known case of isotropic rectangular beam is provided features that are common for all beams are emphasized along with pointing out qualitative differences that exist for beams other than rectangular or circular

a refined use of the residue theorem for the evaluation of band averaged input power into linear second order dynamic systems abstract in a recent work a strategy was proposed which exploits the residue theorem as an efficient tool for evaluating the frequency averaged input power into vibrating systems in this paper such a technique is generalised and further insight and improvements are presented evaluating band averages requires the solution of a weighted integral over a real frequency variable in this paper the integration path is moved to the complex plane where the input mobility shows a smoother behaviour consequently a smaller number of quadrature points are needed to perform an accurate integration which leads to a significant reduction of computational time in addition the connection between the use of quadrature rules in the complex plane and weighting functions on real frequencies is considered two application examples prove the accuracy of the present strategy for different quadrature rules adaptive integration schemes are also investigated

probabilistic properties of the dual structure of the multidimensional knapsack problem and fast statistically efficient algorithms properties of several dual characteristics of the multidimensional knapsack problem such as the probability of existence of optimal and optimal feasible lagrange function generalized saddle points magnitude of relative duality gap etc are investigated for different probabilistic models sufficient conditions of good asymptotic behavior of the dual characteristics are given a fast statistically efficient approximate algorithm with linear running time complexity for problems with random coefficients is presented

computation of propagative waves in free rail using a finite element technique abstract a method is presented for the numerical computation of the wavenumbers and associated modes of the cross section of solid waveguides the method based on the finite element technique is well suited to the computation of both the propagative and the evanescent waves in a straight waveguide with an arbitrary cross section the solution is obtained by factorization with the cross section of the waveguide being modelled by numerical discretization the dispersion curves of propagative waves in a free rail are computed by using triangular and quadrilateral finite elements of the cross section the evolution of cross section modes as the frequency increases is evaluated and discussed

a constrained nonlinear 0 1 program for data allocation abstract this paper analyzes the problem of allocating copies of relations from a global database to the sites of a geographically distributed communication network the objective of the allocation is to minimize the total cost due to transmissions generated by queries from the various sites including queries that access multiple relations this allocation problem is modeled as a constrained nonlinear 0 1 subproblems generated during subgradient optimization are solved as optimization some of the unconstrained quadratic 0 1 subproblems generated during subgradient optimization are solved as maximum flow problems while the others require implicit enumeration depending on the nature of the objective function coefficients of the subproblems our solution approach is tested extensively on data allocation problems with as many as 100 sites and 20 relations on a set of randomly generated test problems our approach was close to two orders of magnitude faster than the general purpose integer programming code osl

a lagrangean relaxation approach for very large scale capacitated lot sizing in this paper we develop a lagrangean relaxation based heuristic procedure to generate near optimal solutions to very large scale capacitated lot sizing problems clsp with setup times and limited overtime our computational results show that large problems involving several thousand products and several thousand 0 1 integer variables can be solved in a reasonable amount of computer time to within one percent of their optimal solution the proposed procedure is general enough to be applied directly or with slight modification to real life production problems

capacitated multi item lot sizing problems with time windows this research concerns a new family of capacitated multi item lot sizing problems namely lot sizing problems with time windows two classes of the problem are analyzed and solved using different lagrangian heuristics capacity constraints and a subset of time window constraints are relaxed resulting in particular single item time window problems that are solved in polynomial time other relaxations leading to the classical wagner whitin problem are also tested several smoothing heuristics are implemented and tested and their results are compared the gaps between lower and upper bounds for most problems are very small less than 1 moreover the proposed algorithms are robust and do not seem to be too affected when different parameters of the problem are varied

performance evaluation of independent superbasic sets on nonlinear replicated networks abstract in this paper we describe a new type of network flow problems that basically consists of the classical transshipment problem with the following extensions 1 the replication of a network by producing subnetworks with identical structure being linked by so called linking arcs 2 the objective function terms related to the linking arcs are non differentiable nonlinear functions by using a networks specialized implementation of a linearly constrained nonlinear programming algorithm described elsewhere we report the computational performance of the new concept of independent superbasic sets that allows to obtain in parallel independent pieces of the solution at each iteration

stronger k tree relaxations for the vehicle routing problem abstract a lagrangian based exact solution algorithm for the vehicle routing problem vrp defined on an undirected graph is introduced in this paper lower bounds are obtained by allowing exponentially many inequalities as candidates to lagrangian dualization three different families of strong valid inequalities each one with exponentially many elements are used within vrp formulations for each of them separation procedures are proposed for points that define incidence vectors of k trees violated inequalities identified in this way are then dualized in a relax and cut framework upper bounds are generated through a lagrangian clarke and wright heuristic a variable fixation test based on approximating linear programming reduced costs is also implemented computational results are presented for the proposed algorithm

a heuristic method for the set covering problem we present a lagrangian based heuristic for the well known set covering problem scp the algorithm was initially designed for solving very large scale scp instances involving up to 5 000 rows an

error bound and reduced gradient projection algorithms for convex minimization over a polyhedral set consider the problem of minimizing over a polyhedral set the composition of an affine mapping with a strongly convex differentiable function the polyhedral set is expressed as the intersection of an affine set with a simpler polyhedral set and a new local error bound for this problem based on projecting the reduced gradient associated with the affine set onto the simpler polyhedral set is studied a class of reduced gradient projection algorithms for solving the case where the simpler polyhedral set is a box is proposed and this bound is used to show that algorithms in this class attain a linear rate of convergence included in this class are the gradient projection algorithm of goldstein and levitin and poljak and an algorithm of bertsekas a new algorithm in this class reminiscent of active set algorithms is also proposed some of the results presented here extend to problems where the objective function is extended real valued and to variational inequality problems

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

causality and mathematical models in vibration and acoustics a realistic perspective causality principles are defined in the context of linear systems with time independent properties the primary concepts are those of a reponse function and of a transfer function the existence of the latter strictly requires that the system be stable and a causal response function can exist without the system being stable the definition of the transfer function can be extended to the negative real axis and subsequently to the upper half plane for specific mathematical models of vibrating systems stability and consequently the use of causality considerations requires that there be no poles in the upper half plane examples are given for a one degree of freedom vibrating system with various types of damping terms negative damping allows the existence of a causal response function but leads to an instable sytem fractional derivative damping is all right if the fractional power is less than 2 consideration of arbitrary multi degree of freedom systems shows that the usual causality considerations apply if the mass damping and stiffness matrices are all positive definite an extended class of kramers kronig relations esists and the general relations in this class are derived which relate the real and imaginary parts of the transfer function

convergence of gradient projection routing methods in an asynchronous stochastic quasi static virtual circuit network the convergence of the gradient projection algorithms for optimal routing in virtual circuit data networks proposed by d p bertsekas 1982 is studied the routing model explicitly takes into account stochastic generation and termination of virtual circuits distributed asynchronous routing updates inaccurate flow measurement and delays in forwarding control packets the problem of assigning paths for incoming sessions or virtual circuits to implement the gradient projection algorithms is also studied a metering rule based on deficiency in a desired number of virtual circuits is proposed and analyzed it is shown that the proposed metering rule is better than a randomized rule in some sense the gradient projection routing algorithms implemented either by the metering rule or the randomized rule are shown to converge to a neighborhood of a long term optimal routing

mode count and modal density of structural systems relationships with boundary conditions the effects of boundary conditions on the mode count and modal density of one and two dimensional structural systems beams and plates respectively are investigated by using the wavenumber integration method bending vibrations are examined first for a single beam in this case it is demonstrated that the average mode count is reduced by between 0 and 1 for each boundary constraint depending on the type of boundary conditions for more generalised mass and stiffness constraints a frequency dependent coefficient which need not lie between 0 and 1 is obtained the effects of line constraints on the mode count of two dimensional systems are similar to the equivalent one dimensional constraints but they are always frequency dependent then the mode count of systems of multiple collinear beams and coplanar plates is studied it is found that an intermediate constraint has the same effect on the average mode count as the same type of constraint applied at an end of the system the modal density is largely independent of boundary conditions for one dimensional systems although there are exceptions while it is dependent on boundary conditions for two dimensional systems the results are compared with those from previously published formulae for natural frequencies and with results from finite element method fem analysis inclusion of the effect of the boundary conditions in statistical energy analysis sea estimations will result in improved agreements with both analytical and numerical results

data parallel quadratic programming on box constrained problems we develop designs for the data parallel solution of quadratic programming problems subject to box constraints in particular we consider the class of algorithms that iterate between projection steps that identify candidate active sets and conjugate gradient steps that explore the working space using the algorithm of more and toraldo report mcs p77 05 89 argonne national laboratory illinois 1989 as a specific instance of this class of algorithms we show how its components can be implemented efficiently on a data parallel simd computer architecture alternative designs are developed for both arbitrary unstructured hessian matrices and for structured problems implementations are carried out on a connection machine cm 2 they are shown to be very efficient achieving a peak computing rate over 2 gflops problems with several hundred thousand variables are solved within one minute of solution time on the 8k cm 2 extremely large test problems with up to 2 89 million variables are also solved efficie

on the use of the residue theorem for the efficient evaluation of band averaged input power into linear second order dynamic systems abstract an alternative to numerical quadrature is proposed to compute the power injected into a vibrating structure over a certain frequency band instead of evaluating the system response at several sampling frequencies within the considered band the integral is computed by estimating the residue at a few complex frequencies corresponding to the poles of the weighting function this technique provides considerable benefits in terms of computation time since the integration is independent of the width of the frequency band two application examples show the effectiveness of the approach firstly the use of a butterworth filter instead of a rectangular weighting function is assessed secondly the accuracy of the approximation in case of hysteretic damping is proven finally the computational performance of the technique is compared with classical numerical quadrature schemes

a branch and bound algorithm for the total weighted tardiness problem this paper presents a new branch and bound algorithm for the single machine total weighted tardiness problem it obtains lower bounds using a lagrangian relaxation approach with subproblems that are total weighted completion time problems the well known subgradient optimization technique is replaced by a multiplier adjustment method that leads to an extremely fast bound calculation the method incorporates various devices for checking dynamic programming dominance in the search tree extensive computational results for problems with up to 50 jobs show the superiority of the algorithm over existing methods

an efficient 3 d stochastic hie fdtd algorithm for investigation of statistical variation in electromagnetic field we propose a stochastic hybrid implicit explicit finite difference time domain method s hie fdtd to compute the mean and variance of the electromagnetic em fields using a single simulation given those of the conductivity and permittivity in the computation domain the mean and variance field update equations underlying the proposed method are derived from the field update equations of the traditional deterministic hie fdtd the courant friedrichs lewy condition of the s hie fdtd depends on the spatial discretization sizes only in two dimensions therefore for computation domains with fine geometric features only in the remaining dimension it uses a time step size that is larger than that of fully explicit schemes indeed numerical results demonstrate that the proposed method is faster than the previously developed stochastic fdtd in computing the mean and variance of the em fields in two different problems wave propagation through a multilayer human tissue and transmission through a frequency selective surface

exact and approximation algorithms for the operational fixed interval scheduling problem the operational fixed interval scheduling problem ofisp is characterized as the problem of scheduling a number of jobs each with a fixed starting time a fixed finishing time a priority index and a job class the objective is to find an assignment of jobs to machines with maximal total priority the problem is complicated by the restrictions that i each machine can handle only one job at a time ii each machine can handle only jobs from a prespecified subset of all possible job classes and iii preemption is not allowed it follows from the above that ofisp has both the character of a job scheduling problem and the character of an assignment problem in this paper we discuss the occurrence of the problem in practice and we present newly developed exact and approximation algorithms for solving ofisp finally some computational results are shown

optimal dynamic vibration absorber design for minimizing the band averaged input power using the residue theorem abstract this paper deals with an efficient strategy to improve the vibro acoustic behavior of a structure over frequency bands genetic algorithms are used to identify the optimal resonance frequency and location of dynamic vibration absorbers dvas which minimize the band averaged input power into a plate leading to an indirect reduction of the radiated acoustic power and global vibration instead of classic numerical quadrature schemes the residue theorem is used to evaluate the band averaged input power this results into a considerable reduction of computational effort as it requires only few function evaluations at complex frequencies regardless of the analyzed bandwidth the structural response is simulated by using the wave based method wbm besides an increased convergence rate as compared to classical element based techniques the wbm is also free in determining the optimal position of the dvas not restricting it to nodal grid locations moreover when point connections are taken into account only a small part of the wb matrices needs to be recomputed at each iteration resulting in a strong reduction of the computation time numerical examples illustrate the benefits and the efficiency of the proposed optimization strategy

a matrix generation approach for eigenvalue optimization we study the extension of a column generation technique to nonpolyhedral models in particular we study the problem of minimizing the maximum eigenvalue of an affine combination of symmetric matrices at each step of the algorithm a restricted master problem in the primal space corresponding to the relaxed dual original problem is formed a query point is obtained as an approximate analytic center of a bounded set that contains the optimal solution of the dual problem the original objective function is evaluated at the query point and depending on its differentiability a column or a matrix is added to the restricted master problem we discuss the issues of recovering feasibility after the restricted master problem is updated by a column or a matrix the computational experience of implementing the algorithm on randomly generated problems are reported and the cpu time of the matrix generation algorithm is compared with that of the primal dual interior point methods on dense and sparse problems using the software sdpt3 our numerical results illustrate that the matrix generation algorithm outperforms primal dual interior point methods on dense problems with no structure and also on a class of sparse problems

an efficient algorithm for the lagrangean dual of nonlinear knapsack problems with additional nested constraints abstract this paper presents an efficient algorithm for solving the lagrangean dual of nonlinear knapsack problems with additional nested constraints the dual solution provides a feasible primal solution if it exists and associated lower and upper bounds on the optimal objective function value of the primal problem computational experience is cited indicating computation time number of dual iterations and tightness of the bounds

energy flow models from finite element analysis computationally efficient methods are described by which the results of a finite element analysis of a system may be post processed to form energy flow models yielding time and perhaps frequency average subsystem energies as well as input and dissipated powers the methods are particularly efficient for excitation which is spatially distributed or broadband e g rain on the roof or if the frequency average response is required first a method based on a global finite element analysis is presented this involves a global modal decomposition and a reordering of the subsequent numerical calculations the properties of the distribution of the excitation and the system s mass and stiffness lead to subsystem force distribution mass distribution and stiffness distribution matrices the response is given by a sum of terms involving the interaction of a pair of global modes the contribution of each pair depending on the appropriate elements of the distribution matrices frequency averaging is performed by separating the resulting frequency dependent terms and integrating in most practical cases this integration can be done analytically next an alternative method involving component mode synthesis is described in this individual finite element analyses are performed for each subsystem using here the fixed interface method these are then assembled to perform a global modal analysis with the order of the model being much reduced the consequent results are then post processed in the same way finally a system comprising three coupled plates is presented as a numerical example

global and asymptotic convergence rate estimates for a class of projected gradient processes projected gradient processes of the goldstein levitin polyak type are considered for constrained minimization problems min omega f with omega a convex set in a hilbert space x and f x to mathbb r 1 a differentiable functional global and local convergence theorems are established for a large class of these processes including those generated with implicit step length rules proposed by bertsekas and goldstein in this analysis traditional uniform strong positivity conditions on the hessian nabla 2 f are replaced by weaker pseudoconvexity conditions and growth conditions on f when f has a unique minimizes in omega convergence rates are shown to depend on how rapidly the function gamma sigma inf r f x f xi mid x in omega x xi geqq sigma grows with increasing sigma 0 if gamma sigma geqq b sigma nu for some b 0 the processes f n in question converge to inf f like o n nu nu 2 linear

accpm with a nonlinear constraint and an active set strategy to solve nonlinear multicommodity flow problems this paper proposes an implementation of a constrained analytic center cutting plane method to solve nonlinear multicommodity flow problems the new approach exploits the property that the objective of the lagrangian dual problem has a smooth component with second order derivatives readily available in closed form the cutting planes issued from the nonsmooth component and the epigraph set of the smooth component form a localization set that is endowed with a self concordant augmented barrier our implementation uses an approximate analytic center associated with that barrier to query the oracle of the nonsmooth component the paper also proposes an approximation scheme for the original objective an active set strategy can be applied to the transformed problem it reduces the dimension of the dual space and accelerates computations the new approach solves huge instances with high accuracy the method is compared to alternative approaches proposed in the literature

a method for the cutting stock problem with different qualities abstract an operations research project at a swedish company is reported the project deals with the problem of allocating wood products with different quality requirements to be cut from wooden boards which in turn consists of a number of quality zones a major difficulty is the existence of distinct time limits since the cross cutting of the boards is done in a real time process in this paper we derive a mathematical model which includes a set packing problem that accurately describes the real world problem this problem is then solved using a combination of lagrangean relaxation and subgradient optimization numerical results for practical problems are reported the results indicate a valid model and satisfactory efficiency of the proposed solution procedure

facilities location in a competitive environment a promethee based multiple criteria analysis abstract in this paper a methodology for locating facilities in a competitive environment is introduced in this methodology the multi criteria approach and the notion of efficiency are appropriately integrated the classic distance criterion is combined with a second one a quality criterion to produce a more realistic representation of a customer s decision process

an active constraints newton algorithm for the spatial price equilibrium problem abstract we consider the general spatial price equilibrium problem in order to solve such problems we develop a newton type algorithm that is combined with an active constraints strategy to handle the nonnegativety constraints or upper bounds on the variables at each iteration of the algorithm the choice of the initial active set is based on the value of the gradient and the dual variables two variants of the algorithm are presented one in which the newton equations are solved by the gauss method and one in which these equations are solved by the conjugate gradient method computational results demonstrate the efficiency of the method

exact finite elements for wave propagation in range independent fluid solid media abstract a finite element technique fem is proposed for solving the two point ordinary differential equation ode boundary value problem for the wavefield in a layered horizontally homogeneous fluid solid medium at fixed horizontal wavenumber k the medium may consist of an arbitrary mixture of fluid and solid layers bounded at both ends by homogeneous half spaces in homogeneous layers the fem basis functions are chosen to be exact local solutions of the governing source free ode s and in a medium composed of homogeneous layers only the overall solution at fixed horizontal wavenumber k is exact without subdivision of the layers in layers with non constant material parameters standard second order accurate linear elements are used the fem stiffness matrix is symmetric with dimension and bandwidth essentially half of those obtained with a parallel shooting global matrix technique the stiffness matrix remains bounded and well conditioned as the lengths of exact elements grow and it can in general be decomposed into triangular factors without scaling or pivoting which contributes significantly to computational efficiency the use of the finite element technique for wavefield synthesis by hankel transform integration or by normal mode expansion is discussed in both cases general purpose numerical quadrature and zero finding techniques can be used if proper care is taken of the poles of the stiffness matrix in the complex k plane a pole free dispersion function efficiently computable from the fem stiffness matrix is derived a simple but useful general result in the theory of non linear eigenvalue problems for matrices is pointed out and used for computing the excitation coefficients of the modes of the discretized system the usefulness of the method is illustrated by presenting numerical results from a computationally intensive application example

using confidence limits for the global optimum in combinatorial optimization let z be the unknown optimal value of a combinatorial optimization problem in minimization form and let z be a constant satisfying probz z 1 for a fixed 0 1 then z is called a statistical or probabilistic lower bound at significance level 1 we discuss how statistical lower bounds can be used to measure the effectiveness of an approximate solution and to fathom subproblems in a branch and bound approach after reviewing some methods for obtaining statistical lower bounds we report on extensive computational experience for two notoriously difficult combinatorial optimization problems the traveling salesman problem and the quadratic assignment problem from our experience we conclude that statistical bounds are highly useful for quadratic assignment problems in this problem context the quality of simple analytic formulas is comparable to the quality of the maximum likelihood formula in general using statistics to aid in the solution of combinatorial optimization problems should be most useful when such problems are large

capacitated lot sizing and scheduling by lagrangean relaxation abstract development of new models and solution procedures for production planning has been of research interest for several decades implementation of these models has resulted in lower production costs by reducing inventories number of setups and labor costs in this paper we develop several optimal near optimal procedures for the capacitated lot sizing and scheduling problem clsp with setup times limited regular time and limited overtime we formulate a mixed integer linear programming model of the problem and solve it by lagrangean relaxation we experiment with alternative lagrangean relaxations and develop new procedures to solve these relaxations overall the capacity constraints relaxation seems to be superior to the demand constraints relaxation our results show that large problems can be solved in reasonable computer times and within one percent accuracy of the optimal solutions we solved 99 8 i e 99 items and 8 periods 50 12 and 50 8 problems in 30 61 36 25 and 12 65 seconds of cdc cyber 730 computer time respectively our procedures are general enough to be applied directly or with slight modifications in real life production settings

stochastic fdtd for analysis of statistical variation in electromagnetic fields this paper describes a new stochastic finite difference time domain s fdtd method for calculating the variance in the electromagnetic fields caused by variability or uncertainty in the electrical properties of the materials in the model details of the 1d derivation using truncated taylor series approximations are given the s fdtd analysis is then compared to monte carlo analysis for a 1d bioelectromagnetic example the accuracy of the method is controlled by the approximations for the cross correlations of the fields and electrical properties in this paper we are able to bound the variance using cross correlations of 1 overestimate and the reflection coefficient underestimate

accelerated dual descent for network flow optimization we present a fast distributed solution to the convex network flow optimization problem our approach uses a family of dual descent algorithms that approximate the newton direction to achieve faster convergence rates than existing distributed methods the approximate newton directions are obtained through matrix splitting techniques and sparse taylor approximations of the inverse hessian we couple this descent direction with a distributed line search algorithm which requires the same information as our descent direction to compute we show that similarly to conventional newton methods the proposed algorithm exhibits super linear convergence within a neighborhood of the optimal value numerical experiments corroborate that convergence times are between one to two orders of magnitude faster than existing distributed optimization methods a connection with recent developments that use consensus to compute approximate newton directions is also presented

dynamic aggregation of set partitioning constraints in column generation column generation is often used to solve problems involving set partitioning constraints such as vehicle routing and crew scheduling problems when these constraints are in large numbers and the columns have on average more than 8 12 nonzero elements column generation often becomes inefficient because solving the master problem requires very long solution times at each iteration due to high degeneracy to overcome this difficulty we introduce a dynamic constraint aggregation method that reduces the number of set partitioning constraints in the master problem by aggregating some of them according to an equivalence relation to guarantee optimality this equivalence relation is updated dynamically throughout the solution process tests on the linear relaxation of the simultaneous vehicle and crew scheduling problem in urban mass transit show that this method significantly reduces the size of the master problem degeneracy and solution times especially for larger problems in fact for an instance involving 1 600 set partitioning constraints the master problem solution time is reduced by a factor of 8

on the use of e most active constraints in an exact penalty function method for nonlinear optimization this note presents an algorithm for nonlinear programming problems which utilizes the e most active constraint strategy in an exact penalty function method with trust region the algorithm is particularly suitable for problems containing a large number of constraints the global convergence of the proposed algorithm is proved the results of limited computational experiments for discretized semi infinite programming problems are also reported to demonstrate the effectiveness of the present approach

conditional subgradient optimization theory and applications abstract we generalize the subgradient optimization method for nondifferentiable convex programming to utilize conditional subgradients firstly we derive the new method and establish its convergence by generalizing convergence results for traditional subgradient optimization secondly we consider a particular choice of conditional subgradients obtained by projections which leads to an easily implementable modification of traditional subgradient optimization schemes to evaluate the subgradient projection method we consider its use in three applications uncapacitated facility location two person zero sum matrix games and multicommodity network flows computational experiments show that the subgradient projection method performs better than traditional subgradient optimization in some cases the difference is considerable these results suggest that our simply modification may improve subgradient optimization schemes significantly this finding is important as such schemes are very popular especially in the context of lagrangean relaxation

partial affine scaling for linearly constrained minimization we propose an interior point method for finding a stationary point of a nonlinear program with linear equality and nonnegativity constraints this method maintains a basis at each iteration and updates the iterate by taking an affine scaling step in the space of nonbasic variables in the general case we propose to choose the basis according to a rule that maximizes the basic components of the iterate in the case where the feasible region is the product of simplices we propose an alternative rule that minimizes the basic components of the cost gradient we analyze the convergence of the method under each rule a key feature of this method is that it can be implemented much like the simplex method

a multiperiod two echelon multicommodity capacitated plant location problem abstract in this paper we deal with a facility location problem where one desires to establish facilities at two different distribution levels by selecting the time periods our model intends to minimize the total cost for meeting demands for all the products specified over the planning horizon at various customer locations while satisfying the capacity requirements of the production plants and intermediate warehouses we address this problem by means of a formulation as a mixed integer programming problem a lagrangean relaxation is proposed to solve the problem together with a heuristic procedure that constructs feasible solutions of the original problem from the solutions at the lower bounds obtained by the relaxed problems computational tests are provided showing the good performance of this approach for a wide range of problems

a logical topology and discrete capacity assignment algorithm for reconfigurable networks a reconfigurable network is a circuit switched network where the effective topology and capacities can be dynamically adapted to changes in the traffic requirements or to changes in the structure of the network due to failures the application of reconfigurable networks considered in this paper is data communications we formulated the joint topology capacity and routing problem in a reconfigurable network as a nonlinear mixed integer programming problem to solve this problem we present a partial branch and bound algorithm the reduced gradient method is used to solve the problem with the integrality constraints relaxed lagrangian relaxation is applied to obtain lower bounds in the computational experiments the algorithm found good solutions in a few minutes of cpu time in addition the reduction in the minimal delay due to the reconfiguration capability can be as large as 62

finding discrete global minima with a filled function for integer programming the filled function method is an approach to find the global minimum of multidimensional functions this paper proposes a new definition of the filled function for integer programming problem a filled function which satisfies this definition is presented furthermore we discuss the properties of the filled function and design a new filled function algorithm numerical experiments on several test problems with up to 50 integer variables have demonstrated the applicability and efficiency of the proposed method

propagating waves and end modes in pretwisted beams abstract a method is presented for studying propagating waves and end modes in a uniformly pretwisted beam the variationally derived equations of motion are based on three dimensional elasticity and finite element modelling of the cross section this discretization procedure accommodates arbitrarily shaped cross sections of inhomogeneous anisotropic material properties that follow the pretwist rotation rate an harmonic solution form in both timetand axial co ordinate is introduced i e exp i k t resulting in a two parameter algebraic eigensystem by specifying the axial wavenumberk the eigenproblem permits real frequencies of propagating modes to be determined by specifying real frequency both real and complex axial wavenumbers can be extracted where real values pertain to propagating modes and the complex ones to edge vibrations or end modes due to pretwist the effect of extension torsion and flexural are coupled examples of homogeneous isotropic beams with rectangular and square cross sections are given to illustrate the method of analysis and the physical behavior

reliable facility location design under the risk of disruptions reliable facility location models consider unexpected failures with site dependent probabilities as well as possible customer reassignment this paper proposes a compact mixed integer program mip formulation and a continuum approximation ca model to study the reliable uncapacitated fixed charge location problem rufl which seeks to minimize initial setup costs and expected transportation costs in normal and failure scenarios the mip determines the optimal facility locations as well as the optimal customer assignments and is solved using a custom designed lagrangian relaxation lr algorithm the ca model predicts the total system cost without details about facility locations and customer assignments and it provides a fast heuristic to find near optimum solutions our computational results show that the lr algorithm is efficient for mid sized rufl problems and that the ca solutions are close to optimal in most of the test instances for large scale problems the ca method is a good alternative to the lr algorithm that avoids prohibitively long running times

modeling and solving the two facility capacitated network loading problem this paper studies a topical and economically significant capacitated network design problem that arises in the telecommunications industry in this problem given point to point communication demand in a network must be met by installing loading capacitated facilities on the arcs loading a facility incurs an arc specific and facility dependent cost this paper develops modeling and solution approaches for loading facilities to satisfy the given demand at minimum cost we consider two approaches for solving the underlying mixed integer program a lagrangian relaxation strategy and a cutting plane approach that uses three classes of valid inequalities that we identify for the problem we show that a linear programming formulation that includes these inequalities always approximates the value of the mixed integer program at least as well as the lagrangian relaxation bound our computational results on a set of prototypical telecommunication data show that including these inequalities considerably improves the gap between the integer programming formulation and its linear programming relaxation from an average of 25 to an average of 8 these results show that strong cutting planes can be an effective modeling and algorithmic tool for solving problems of the size that arise in the telecommunications industry

electromagnetic uncertainty analysis using stochastic fdfd method this paper presents a novel stochastic finite difference frequency domain s fdfd method that incorporates the statistical variations of electrical properties directly into the fdfd algorithm by using the delta method approximation the method is validated through two examples a 1 d bioelectromagnetic layered model and specific absorption rate in a 2 d slice of a human head model comparisons between the single run s fdfd results and those of the monte carlo method the benchmark in statistical simulations show that the s fdfd method can predict the mean and variance of electromagnetic fields with a fair degree of accuracy in a computationally efficient way our numerical experiment also shows that the accuracy is highly dependent on the approximation for the correlation coefficients of the variance equations

projected newton methods for optimization problems with simple constraints we consider the problem min f x x geqq 0 and propose algorithms of the form x k 1 x k alpha k d k nabla f x k where cdot denotes projection on the positive orthant alpha k is a stepsize chosen by an armijo like rule and d k is a positive definite symmetric matrix which is partly diagonal we show that d k can be calculated simply on the basis of second derivatives of f so that the resulting newton like algorithm has a typically superlinear rate of convergence with other choices of d k convergence at a typically linear rate is obtained the algorithms are almost as simple as their unconstrained counterparts they are well suited for problems of large dimension such as those arising in optimal control while being competitive with existing methods for low dimensional problems the effectiveness of the newton like algorithm is demonstrated via computational examples involving as many as 10 000 variables extensions to general linearly constrained pr

input power to waveguides calculated by a finite element method this paper considers the power injected into waveguide structures from forces that are concentrated along the waveguide but are otherwise arbitrary the motion of the structure is described by a set of coupled linear one dimensional wave equations these equations are in this work devised by the waveguide finite element method which is a versatile tool for describing wave motion in general structures that have uniform properties along one direction two separate procedures are derived the first is based on spatial and frequency averaging of a modal solution the second procedure is based on a spatial fourier transform which by nature assumes the waveguide to be infinite it is shown that in the limit of zero damping the two methods are identical subsequently input power to a stiffener in a railway car structure is calculated and compared with an in situ measurement showing a fair agreement

on the power input to point loaded hysteretically damped structures abstract the power input to point loaded hysteretically damped one and two dimensional structures is investigated both uniform and periodic structures are considered and in each case the input power is expressed in terms of the solutions of the damped dispersion relation in the case of a one dimensional periodic structure for example it is found that the input power is directly proportional to the rate of change of the phase constant with frequency an established approximate formula states that the power input to a structure is independent of damping and proportional to the modal density by considering the specific cases of a beam on multiple supports and a cylindrical shell it is shown here that the damping can significantly affect the input power

finite element computation of dispersion properties of thin walled waveguides abstract a method is presented for the numerical computation of the wavenumbers and associated modes of the cross section of thin walled waveguides the method is based on finite element techniques a thin shell type finite element of the cross section is developed by using the virtual work formulation for one dimensional propagation the case of a waveguide the set of equations of motion evaluated by finite element discretization is then transformed into a simple eigenvalue problem involving a general real matrix two examples are analyzed by the finite element method developed the circular cylindrical shell straight thin walled pipe and the thin walled beam of i shaped cross section the so called i profile results obtained by the finite element method by donnell s shell theory and by using simple euler bernoulli beam theory are compared

a unified description of iterative algorithms for traffic equilibria the purpose of this paper is to provide a unified description of iterative algorithms for the solution of traffic equilibrium problems we demonstrate that a large number of well known solution techniques can be described in a unified manner through the concept of partial linearization and establish close relationships with other algorithmic classes for nonlinear programming and variational inequalities in the case of nonseparable travel costs the class of partial linearization algorithms are shown to yield new results in the theory of finite dimensional variational inequalities the possibility of applying truncated algorithms within the framework is also discussed

a new lagrangian relaxation approach to the generalized assignment problem abstract in this paper we present a new lagrangian relaxation approach to the generalized assignment problem gap the new approach is based on a reformulation of gap into an equivalent problem which is then relaxed by traditional lagrangian relaxation techniques the reformulation is created by introducing a set of auxiliary variables and a number of coupling constraints by relaxing the coupling constraints we get subproblems where both types of constraint structures present in the gap are active the lower bound from the lagrangian dual of this approach is shown to be at least as strong as that from the best of the traditional lagrangian relaxation approaches an important feature is also the possibility of generating upper bounds quite easily the new approach has been tested on 10 problems of size m 4 n 25 i e 100 0 1 variables and the results have been compared with the traditional approaches

applications of the causality condition to one dimensional acoustic reflection problems the causality condition is examined as a means of determining frequency domain information about a submerged object from a partial knowledge of its acoustic reflection characteristics a one dimensional problem is considered in which an acoustic wave reflects from an object that is described by the impedance it presents to the fluid two new applications of the causality condition to the frequency domain analysis of this problem are investigated and illustrated by numerical examples in each application the causality condition is used to find the object s complex impedance from a knowledge of the reflected wave s magnitude the first application is to experimental studies where one desires a knowledge of an object s complex impedance but practical limitations only allow a measurement of the reflected wave amplitude analysis shows that the causality condition may be used to determine the phase of the reflected wave and hence the object s impedance if the reflection coefficient is minimum phase when this is true examples suggest that the phase of the reflection coefficient may be accurately determined from the causality condition even in the presence of noise and band limited data the second application is to design situations where one wishes to create an object that reflects sound with a specified frequency dependent magnitude the causality condition may aid the designer by providing a knowledge of all causal object impedances that produce the same reflection coefficient magnitude a numerical example is presented in which a variety of causal object impedances produce the same reflection coefficient magnitude over an infinite frequency range

generating fenchel cutting planes for knapsack polyhedra the author recently proposed a class of cutting planes for integer programs called fenchel cuts which distinguish themselves from more conventional cuts in that they are generated by directly seeking to solve the separation problem rather than by using explicit knowledge of the polyhedral structure of the integer program an algorithm for generating fenchel cuts is presented and described in detail for the separation problem associated with knapsack polyhedra computational results are presented for a collection of real world integer programs to demonstrate the effectiveness of the cutting planes

the tv break packing problem abstract instead of selling advertisement spots one by one some french satellite channels decided in 2002 to modify their commercial offer in order to sell packages of spots these new general conditions of sale lead to an interesting optimization problem that we named the tv b reak p acking p roblem tvbp we establish its np hardness and study various resolutions approaches including linear programming lp lagrangian relaxation lr constraint programming cp and local search ls finally we propose a generic cp ls hybridization scheme branch and move whose application to the tvbp obtained the best results in our experiments dual upper bounds of the maximal revenue are also computed

multiple models multiplicative noise and linear quadratic control algorithmic aspects numerical solution of certain linear quadratic lq control problems for robust design is addressed an iterative method is suggested and analysed for solving a minimax multiple model lq control problem a convergent iterative method is studied for finding the constant feedback gains of a given linear controller so as to solve a spectral radius functional minimization problem for multiple plants furthermore a convergent iterative method is proposed for solving the maximum entropy parametric lq control problem with multiplicative noise

a multiplier adjustment method for the generalized assignment problem we describe a branch and bound algorithm for the generalized assignment problem in which bounds are obtained from a lagrangian relaxation with the multipliers set by a heuristic adjustment method the algorithm was tested on a large sample of small random problems and a number of large problems derived from a vehicle routing application computation times were reasonable in all cases and the branch and bound trees generated had nearly two orders of magnitude fewer nodes than for competing algorithms although comparison of running times on different machines is difficult the multiplier adjustment method appears to be about one order of magnitude faster than the best previously existing algorithms for this problem

single machine scheduling with symmetric earliness and tardiness penalties abstract this research focuses on scheduling jobs with varying processing times and distinct due dates on a single machine subject to earliness and tardiness penalties hence this work will find application in a just in time jit production environment the scheduling problem is formulated as a 0 1 linear integer program with three sets of constraints where the objective is to minimize the sum of the absolute deviations between job completion times and their respective due dates the first two sets of constraints are equivalent to the supply and demand constraints of an assignment problem the third set which represents the process time non overlap constraints is relaxed to form the lagrangian dual problem the dual problem is then solved using the subgradient algorithm efficient heuristics have also been developed in this work to yield initial primal feasible solutions and to convert primal infeasible solutions to feasibility the computational results show that the relative deviation from optimality obtained by the subgradient algorithm is less than 3 for problem sizes varying from 10 to 100 jobs