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

an efficient trajectory method for probabilistic production inventory distribution problems we consider a supply chain operating in an uncertain environment the customers demand is characterized by a discrete probability distribution a probabilistic programming approach is adopted for constructing an inventory production distribution plan over a multiperiod planning horizon the plan does not allow the backlogging of the unsatisfied demand and minimizes the costs of the supply chain while enabling it to reach a prescribed nonstockout service level it is a strategic plan that hedges against undesirable outcomes and that can be adjusted to account for possible favorable realizations of uncertain quantities a modular integrated and computationally tractable method is proposed for the solution of the associated stochastic mixed integer optimization problems containing joint probabilistic constraints with dependent right hand side variables the concept of p efficiency is used to construct a finite number of demand trajectories which in turn are employed to solve problems with joint probabilistic constraints we complement this idea by designing a preordered set based preprocessing algorithm that selects a subset of promising p efficient demand trajectories finally to solve the resulting disjunctive mixed integer programming problem we implement a special column generation algorithm that limits the risk of congestion in the resources of the supply chain the methodology is validated on an industrial problem faced by a large chemical supply chain and turns out to be very efficient it finds a solution with a minimal integrality gap and provides substantial cost savings

facets and algorithms for capacitated lot sizing the dynamic economic lot sizing model which lies at the core of numerous production planning applications is one of the most highly studied models in all of operations research and yet capacitated multi item versions of this problem remain computationally elusive we study the polyhedral structure of an integer programming formulation of a single item capacitated version of this problem and use these results to develop solution methods for multi item applications in particular we introduce a set of valid inequalities for the problem and show that they define facets of the underlying integer programming polyhedron computational results on several single and multiple product examples show that these inequalities can be used quite effectively to develop an efficient cutting plane branch and bound procedure moreover our results show that in many instances adding certain of these inequalities a priori to the problem formulation and avoiding the generation of cutting planes can be equally effective

mapping the orientation of intravoxel crossing fibers based on the phase information of diffusion circular spectrum a new method is presented to map the orientation of intravoxel crossing fibers by using the phase of the diffusion circular spectrum harmonics in a previous study zhan w gu h xu s silbersweig d a stern e yang y 2003 circular spectrum mapping for intravoxel fiber structures based on high angular resolution apparent diffusion coefficients magn reson med 49 1077 1088 we demonstrated that the magnitude of the 4th order harmonic of the diffusion circular spectrum can be used to identify the existence of fiber crossings however the orientation of the intravoxel crossing fibers remained unknown this study extends the diffusion circular spectrum mapping method so that it is able to identify the orientation of the intravoxel crossing fibers by utilizing the phase information of the circular spectrum in general the phase of the circular harmonic determines the rotation of the apparent diffusion coefficient adc profile on the sampling circle that is spanned by the major and medium eigenvector of the diffusion tensor and thus can be used to determine the orientation of the crossing fibers simulation results show that the regular tensor based major eigenvector maps have obvious artifacts in the fiber crossing area whereas the estimated crossing fibers by the proposed method are much more consistent with the orientation of the actual intravoxel fibers diffusion mri experiments were performed on five healthy human brains using a 3t scanner the brain regions with fiber crossings were selected by thresholding the magnitudes of the 4th order circular spectrum map intravoxel crossing fibers were estimated by the phase of the 4th order harmonic for each voxel within these areas the estimated intravoxel crossing fibers demonstrated a clear consistency with the orientations of fiber tracks in the surrounding tissues reducing the fiber orientation discontinuity of the regular major eigenvector map

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

regularized positive definite fourth order tensor field estimation from dw mri in diffusion weighted magnetic resonance image dw mri processing a 2nd order tensor has been commonly used to approximate the diffusivity function at each lattice point of the dw mri data from this tensor approximation one can compute useful scalar quantities e g anisotropy mean diffusivity which have been clinically used for monitoring encephalopathy sclerosis ischemia and other brain disorders it is now well known that this 2nd order tensor approximation fails to capture complex local tissue structures e g crossing fibers and as a result the scalar quantities derived from these tensors are grossly inaccurate at such locations in this paper we employ a 4th order symmetric positive definite spd tensor approximation to represent the diffusivity function and present a novel technique to estimate these tensors from the dw mri data guaranteeing the spd property several articles have been reported in literature on higher order tensor approximations of the diffusivity function but none of them guarantee the positivity of the estimates which is a fundamental constraint since negative values of the diffusivity are not meaningful in this paper we represent the 4th order tensors as ternary quartics and then apply hilbert s theorem on ternary quartics along with the iwasawa parametrization to guarantee an spd 4th order tensor approximation from the dw mri data the performance of this model is depicted on synthetic data as well as real dw mris from a set of excised control and injured rat spinal cords showing accurate estimation of scalar quantities such as generalized anisotropy and trace as well as fiber orientations

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

short term scheduling of thermal electric generators using lagrangian relaxation this paper presents an expanded formulation of the unit commitment problem in which hundreds of thermal electric generators must be scheduled on an hourly basis for up to 7 days at a time the underlying model incorporates the full set of costs and constraints including setup production ramping and operational status and takes the form of a mixed integer nonlinear control problem lagrangian relaxation is used to disaggregate the model by generator into separate subproblems which are then solved with a nested dynamic program the strength of the methodology lies partially in its ability to construct good feasible solutions from information provided by the dual thus the need for branch and bound is eliminated in addition the inclusion of the ramping constraint provides new insight into the limitations of current techniques computational experience with the proposed algorithm indicates that problems containing up to 100 units and 48 time periods can be readily solved in reasonable times duality gaps of less than 1 were achieved in all cases

dt refind diffusion tensor registration with exact finite strain differential in this paper we propose the dt refind algorithm for the diffeomorphic nonlinear registration of diffusion tensor images unlike scalar images deforming tensor images requires choosing both a reorientation strategy and an interpolation scheme current diffusion tensor registration algorithms that use full tensor information face difficulties in computing the differential of the tensor reorientation strategy and consequently these methods often approximate the gradient of the objective function in the case of the finite strain fs reorientation strategy we borrow results from the pose estimation literature in computer vision to derive an analytical gradient of the registration objective function by utilizing the closed form gradient and the velocity field representation of one parameter subgroups of diffeomorphisms the resulting registration algorithm is diffeomorphic and fast we contrast the algorithm with a traditional fs alternative that ignores the reorientation in the gradient computation we show that the exact gradient leads to significantly better registration at the cost of computation time independently of the choice of euclidean or log euclidean interpolation and sum of squared differences dissimilarity measure the exact gradient achieves better alignment over an entire spectrum of deformation penalties alignment quality is assessed with a battery of metrics including tensor overlap fractional anisotropy inverse consistency and closeness to synthetic warps the improvements persist even when a different reorientation scheme preservation of principal directions is used to apply the final deformations

characterization of a class of distance regular graphs on demontre que tout graphe distance regulier de diametre d 3 avec un certain reseau d intersections est isomorphe a 1 des graphes suivants des graphes de hamming graphe d egawa graphe de johnson demi cubes graphe de gosset

scheduling of manufacturing systems using the lagrangian relaxation technique the practical solutions for three manufacturing scheduling problems are examined as each problem is formulated constraints are added or modified to reflect increasing real world complexity the first problem considers scheduling single operation jobs on identical machines the second problem is concerned with scheduling multiple operation jobs with simple fork join precedence constraints on identical machines the third problem is the job shop problem in which multiple operation jobs with general precedence constraints are scheduled on multiple machine types langrangian relaxation is used to decompose each of the scheduling problems into job or operation level subproblems the subproblems are easier to solve than the original problem and have intuitive appeal this technique results in algorithms which generate near optimal schedules efficiently while giving a lower bound on the optimal cost in resolving the scheduling problem from one time instant to the next the lagrange multipliers from the last schedule can be used to initialize the multipliers further reducing the computation time

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

computing large deformation metric mappings via geodesic flows of diffeomorphisms this paper examine the euler lagrange equations for the solution of the large deformation diffeomorphic metric mapping problem studied in dupuis et al 1998 and trouve 1995 in which two images i 0 i 1 are given and connected via the diffeomorphic change of coordinates i 0 1 i 1 where 1 is the end point at t 1 of curve t t 0 1 satisfying t v t t t 0 1 with 0 id the variational problem takes the form mathop arg text m in limits upsilon dot phi t upsilon t left dot phi right left int 0 1 left upsilon t right 2 text d t left i 0 circ phi 1 1 i 1 right l 2 2 right where v t v is an appropriate sobolev norm on the velocity field v t and the second term enforces matching of the images with l 2 representing the squared error norm in this paper we derive the euler lagrange equations characterizing the minimizing vector fields v t t 0 1 assuming sufficient smoothness of the norm to guarantee existence of solutions in the space of diffeomorphisms we describe the implementation of the euler equations using semi lagrangian method of computing particle flows and show the solutions for various examples as well we compute the metric distance on several anatomical configurations as measured by 0 1 v t v dt on the geodesic shortest paths

exceptional graphs with smallest eigenvalue 2 and related problems this paper summarizes the known results on graphs with smallest eigenvalue around 2 and completes the theory by proving a number of new results giving comprehensive tables of the finitely many exceptions and posing some new problems then the theory is applied to characterize a class of distance regular graphs of large diameter by their intersection array

hierarchical design of an integrated production and 2 echelon distribution system abstract this study uses the method of lagrangean relaxation in the hierarchical design of an integrated model of production distribution functions in a 2 echelon system a mixed integer mathematical model is developed with a centralized planning perspective to address production and distribution decisions simultaneously in order to solve the resulting large scale problem the lagrangean relaxation is used to decouple the imbedded distribution and production subproblems and subgradient optimization is implemented to coordinate the information flow between these in a hierarchical manner this corresponds to a decentralized organizational design where a central agent coordinates the information exchange between the distribution and production organizational units a forward heuristic designed to solve the distribution subproblem is shown to provide good solutions hierarchical interdependency is incorporated into the lagrangean heuristic such that distribution decisions are placed in the top level to restrict the solution of the production subproblem in the lower level

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

using lagrangean techniques to solve hierarchical production planning problems this paper proposes and tests a procedure for decomposing a large scale production planning problem modeled as a mixed integer linear program we interpret this decomposition in the context of hax and meal s hierarchical framework for production planning the procedure decomposes the production planning problem into two subproblems which correspond to the aggregate planning subproblem and a disaggregation subproblem in the hax meal framework the linking mechanism for these two subproblems is an inventory consistency relationship which is priced out by a set of lagrange multipliers the best values for the multipliers are found by an iterative procedure which may be interpreted as a feedback mechanism in the hax meal framework at each iteration the procedure finds both a lower bound on the optimal value to the production planning problem and a feasible solution from which an upper bound is obtained our computational tests show that the best feasible solution found from this procedure is very close to o

fluid registration of diffusion tensor images using information theory we apply an information theoretic cost metric the symmetrized kullback leibler skl divergence or j divergence to fluid registration of diffusion tensor images the difference between diffusion tensors is quantified based on the skl divergence of their associated probability density functions pdfs three dimensional dti data from 34 subjects were fluidly registered to an optimized target image to allow large image deformations but preserve image topology we regularized the flow with a large deformation diffeomorphic mapping based on the kinematics of a navier stokes fluid a driving force was developed to minimize the j divergence between the deforming source and target diffusion functions while reorienting the flowing tensors to preserve fiber topography in initial experiments we showed that the skl divergence based on full diffusion pdfs is adaptable to higher order diffusion models such as high angular resolution diffusion imaging hardi the skl divergence was sensitive to subtle differences between two diffusivity profiles showing promise for nonlinear registration applications and multisubject statistical analysis of hardi data

a benders decomposition based heuristic for the hierarchical production planning problem abstract in this paper a heuristic procedure for determining good feasible solutions to a multi item dynamic production planning problem is described and evaluated the problem which is modelled as a mixed integer linear program has a hierarchical structure where production items are aggregated into families and families into product types a decomposition of the original problem creates a trivial subproblem providing feasible solutions on the type level and a master problem which separates into one uncapacitated lot sizing problem for each family the latter problems can be solved by dynamic programming with very little computational effort besides the nice structure of both the sub and master problem the dual information from the subproblem can be given interesting economic interpretations as distinguished from hax and meal s hierarchical framework for production planning and also they hybrid approach suggested by graves in 1982 there exists a direct information flow from the family level problem to the type level problem in form of a production plan the heuristic is essentially an application of benders decomposition the only difference being that the benders cuts are relaxed using lagrangean multipliers a subgradient procedure is used to update the lagrangean multipliers one iteration of this heuristic requires less computational effort compared to grave s algorithm this is due to the fact that the benders subproblem can be solved by inspection while graves algorithm involves solving an lp problem on the type level for thirtysix medium size test problems the average deviation from optimum is 2 34 and the deviation ranges between 0 00 and 5 95

deformable templates using large deformation kinematics a general automatic approach is presented for accommodating local shape variation when mapping a two dimensional 2 d or three dimensional 3 d template image into alignment with a topologically similar target image local shape variability is accommodated by applying a vector field transformation to the underlying material coordinate system of the template while constraining the transformation to be smooth globally positive definite jacobian smoothness is guaranteed without specifically penalizing large magnitude deformations of small subvolumes by constraining the transformation on the basis of a stokesian limit of the fluid dynamical navier stokes equations this differs fundamentally from quadratic penalty methods such as those based on linearized elasticity or thin plate splines in that stress restraining the motion relaxes over time allowing large magnitude deformations kinematic nonlinearities are inherently necessary to maintain continuity of structures during large magnitude deformations and are included in all results after initial global registration final mappings are obtained by numerically solving a set of nonlinear partial differential equations associated with the constrained optimization problem automatic regridding is performed by propagating templates as the nonlinear transformations evaluated on a finite lattice become singular application of the method to intersubject registration of neuroanatomical structures illustrates the ability to account for local anatomical variability

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

optimization based production planning with hybrid dynamics and constraints optimizing production planning has tremendous economic impact for many industrial production systems in this paper the planning problem of a class of production systems with hybrid dynamics and constraints is considered with practical background of power generation planning and other applications the problem is solved within the lagrangian relaxation framework with the system wide demand and resource limit constraints relaxed by lagrange multipliers a new method is developed in this paper to obtain the exact optimal solutions to the subproblems with hybrid dynamics and constraints efficiently without discretizing the continuous production levels or introducing intermediate levels of relaxation a novel definition of the discrete state associated with a consecutive time span is introduced so that solving each subproblem is converted into solving a number of continuous optimization problems and a discrete optimization problem separately an efficient double dynamic programming dp method is developed to solve these subproblems and the principle of optimality is guaranteed for both the continuous and discrete problem the production levels in a consecutive running span with non convex piecewise linear cost functions are determined in a dp forward sweep without discretization the dp method is then applied to determine the optimal discrete operating states across time efficiently numerical testing results demonstrate that the new method is efficient and effective for optimization based production planning with the complex hybrid dynamics and constraints

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

diffeomorphic image registration of diffusion mri using spherical harmonics nonrigid registration of diffusion magnetic resonance imaging mri is crucial for group analyses and building white matter and fiber tract atlases most current diffusion mri registration techniques are limited to the alignment of diffusion tensor imaging dti data we propose a novel diffeomorphic registration method for high angular resolution diffusion images by mapping their orientation distribution functions odfs odfs can be reconstructed using q ball imaging qbi techniques and represented by spherical harmonics shs to resolve intra voxel fiber crossings the registration is based on optimizing a diffeomorphic demons cost function unlike scalar images deforming odf maps requires odf reorientation to maintain its consistency with the local fiber orientations our method simultaneously reorients the odfs by computing a wigner rotation matrix at each voxel and applies it to the sh coefficients during registration rotation of the coefficients avoids the estimation of principal directions which has no analytical solution and is time consuming the proposed method was validated on both simulated and real data sets with various metrics which include the distance between the estimated and simulated transformation fields the standard deviation of the general fractional anisotropy and the directional consistency of the deformed and reference images the registration performance using shs with different maximum orders were compared using these metrics results show that the diffeomorphic registration improved the affine alignment and registration using shs with higher order shs further improved the registration accuracy by reducing the shape difference and improving the directional consistency of the registered and reference odf maps

timer tensor image morphing for elastic registration we propose a novel diffusion tensor imaging dti registration algorithm called tensor image morphing for elastic registration timer which leverages the hierarchical guidance of regional distributions and local boundaries both extracted directly from the tensors currently available dti registration methods generally extract tensor scalar features from each tensor to construct scalar maps subsequently regional integration and other operations such as edge detection are performed to extract more features to guide the registration however there are two major limitations with these approaches first the computed regional features might not reflect the actual regional tensor distributions second by the same token gradient maps calculated from the tensor derived scalar feature maps might not represent the actual tissue tensor boundaries to overcome these limitations we propose a new approach which extracts regional and edge information directly from a tensor neighborhood regional tensor distribution information such as mean and variance is computed in a multiscale fashion directly from the tensors by taking into account the voxel neighborhood of different sizes and hence capturing tensor information at different scales which in turn can be employed to hierarchically guide the registration such multiscale scheme can help alleviate the problem of local minimum and is also more robust to noise since one can better determine the statistical properties of each voxel by taking into account the properties of its surrounding also incorporated in our method is edge information extracted directly from the tensors which is crucial to facilitate registration of tissue boundaries experiments involving real subjects simulated subjects fiber tracking and atrophy detection indicate that timer performs better than the other methods yang et al 2008 zhang et al 2006

solving mixed integer programming problems using automatic reformulation in this paper we describe computational experience in solving mixed 0 1 programming problems using strong valid inequalities as cutting planes in particular we report on the solution to optimality of 18 medium to large size problems including production planning problems with setup costs and capacity constraints multilevel distribution planning problems drainage and heating system design problems and electricity generator scheduling problems the solution approach uses the theory of strong valid inequalities that we developed in a series of earlier papers here we report specifically on the implementation of an experimental system mpsarx which consists of the sciconic mathematical programming system and an automatic reformulation executor arx that use this theory and on the results obtained with this system

solving the fixed charge problem with lagrangian relaxation and cost allocation heuristics abstract a new heuristic algorithm is presented for solving large general fixed charge problems based on langrangian relaxation and cost allocation heuristics limited computational results indicate the new algorithm provides good solutions to large problems with reasonable computational requirements it demonstrates significant improvement over current methods for solving large general fixed charge problems

a mathematical programming model and solution for scheduling production orders in shanghai baoshan iron and steel complex in this paper we investigate the production order scheduling problem derived from the production of steel sheets in shanghai baoshan iron and steel complex baosteel a deterministic mixed integer programming mip model for scheduling production orders on some critical and bottleneck operations in baosteel is presented in which practical technological constraints have been considered the objective is to determine the starting and ending times of production orders on corresponding operations under capacity constraints for minimizing the sum of weighted completion times of all orders due to large numbers of variables and constraints in the model a decomposition solution methodology based on a synergistic combination of lagrangian relaxation linear programming and heuristics is developed unlike the commonly used method of relaxing capacity constraints this methodology alternatively relaxes constraints coupling integer variables with continuous variables which are introduced to the objective function by lagrangian multipliers the lagrangian relaxed problem can be decomposed into two sub problems by separating continuous variables from integer ones the sub problem that relates to continuous variables is a linear programming problem which can be solved using standard software package osl while the other sub problem is an integer programming problem which can be solved optimally by further decomposition the subgradient optimization method is used to update lagrangian multipliers a production order scheduling simulation system for baosteel is developed by embedding the above lagrangian heuristics computational results for problems with up to 100 orders show that the proposed lagrangian relaxation method is stable and can find good solutions within a reasonable time

dual applications of proximal bundle methods including lagrangian relaxation of nonconvex problems we exhibit useful properties of proximal bundle methods for finding min sf where f and s are convex we show that they asymptotically find objective subgradients and constraint multipliers involved in optimality conditions multipliers of objective pieces for max type functions and primal and dual solutions in lagrangian decomposition of convex programs when applied to lagrangian relaxation of nonconvex programs they find solutions to relaxed convexified versions of such programs numerical results are presented for unit commitment in power production scheduling

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

optimal release times in a single server an optimal control perspective the paper is concerned with the basic structure of optimal control of discrete event dynamic processes defined over a max plus algebra only a simple system is considered namely a single server processing a given sequence of jobs but the structural conditions that are discovered may lead to extensions to more general systems the problem in question is how to optimally control the completion output times of the jobs by assigning their release input times so as to minimize a measure of the discrepancy between the completion times and given desired due dates the concept of the costate is applied to the discrete dynamics to identify structural optimality conditions and in the case of quadratic cost measures the optimal control is shown to be computable by a state feedback law that is linear in the max plus algebra

a large deformation diffeomorphic metric mapping solution for diffusion spectrum imaging datasets spatial transformation for diffusion spectrum imaging dsi is an important step for group analyses of dsi datasets in this study we developed a transformation method for dsi datasets under the framework of large deformation diffeomorphic metric mapping lddmm which is termed lddmm dsi the proposed method made use of the fact that a dsi dataset is 6d and generalized the original 2d 3d lddmm algorithm to the 6d case with some modifications made for the dsi datasets in this manner the conventional reorientation problem that arises from transforming diffusion weighted datasets was avoided by making the dsi datasets capable of being freely deformed in the q space the algorithm treated the data matching task as a variational problem under the lddmm framework and sought optimal velocity fields from which the generated transformations were diffeomorphic and the transformation curve was a geodesic the mathematical materials and numerical implementation are detailed in the paper and experiments were performed to analyze the proposed method on real brain dsi datasets the results showed that the method was capable of registering different dsi datasets in both global structural shapes and local diffusion profiles in conclusion the proposed method can facilitate group analyses of dsi datasets and the generation of a dsi template

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

the terwilliger polynomial of a q polynomial distance regular graph and its application to pseudo partition graphs abstract let be a q polynomial distance regular graph with diameter at least 3 terwilliger 1993 implicitly showed that there exists a polynomial say t r of degree 4 depending only on the intersection numbers of and its q polynomial ordering and such that t 0 holds for any non principal eigenvalue of the local graph x for any vertex x v we call t the terwilliger polynomial of in this paper we give an explicit formula for t in terms of the intersection numbers of and the dual eigenvalues of with respect to the first primitive idempotent in its q polynomial ordering we then apply this polynomial to show that all pseudo partition graphs with diameter at least 3 are known

high dimensional spatial normalization of diffusion tensor images improves the detection of white matter differences an example study using amyotrophic lateral sclerosis spatial normalization of diffusion tensor images plays a key role in voxel based analysis of white matter wm group differences currently it has been achieved using low dimensional registration methods in the large majority of clinical studies this paper aims to motivate the use of high dimensional normalization approaches by generating evidence of their impact on the findings of such studies using an ongoing amyotrophic lateral sclerosis als study we evaluated three normalization methods representing the current range of available approaches low dimensional normalization using the fractional anisotropy fa high dimensional normalization using the fa and high dimensional normalization using full tensor information each method was assessed in terms of its ability to detect significant differences between als patients and controls our findings suggest that inadequate normalization with low dimensional approaches can result in insufficient removal of shape differences which in turn can confound fa differences in a complex manner and that utilizing high dimensional normalization can both significantly minimize the confounding effect of shape differences to fa differences and provide a more complete description of wm differences in terms of both size and tissue architecture differences we also found that high dimensional approaches by leveraging full tensor features instead of tensor derived indices can further improve the alignment of wm tracts

diffeomorphic metric mapping of high angular resolution diffusion imaging based on riemannian structure of orientation distribution functions in this paper we propose a novel large deformation diffeomorphic registration algorithm to align high angular resolution diffusion images hardi characterized by orientation distribution functions odfs our proposed algorithm seeks an optimal diffeomorphism of large deformation between two odf fields in a spatial volume domain and at the same time locally reorients an odf in a manner such that it remains consistent with the surrounding anatomical structure to this end we first review the riemannian manifold of odfs we then define the reorientation of an odf when an affine transformation is applied and subsequently define the diffeomorphic group action to be applied on the odf based on this reorientation we incorporate the riemannian metric of odfs for quantifying the similarity of two hardi images into a variational problem defined under the large deformation diffeomorphic metric mapping framework we finally derive the gradient of the cost function in both riemannian spaces of diffeomorphisms and the odfs and present its numerical implementation both synthetic and real brain hardi data are used to illustrate the performance of our registration algorithm

progressive interval heuristics for multi item capacitated lot sizing problems we consider a family of n items that are produced in or obtained from the same production facility demands are deterministic for each item and each period within a given horizon of t periods if in a given period an order is placed setup costs are incurred the aggregate order size is constrained by a capacity limit the objective is to find a lot sizing strategy that satisfies the demands for all items over the entire horizon without backlogging and that minimizes the sum of inventory carrying costs fixed order costs and variable order costs all demands cost parameters and capacity limits may be time dependent in the basic joint setup cost js model the setup cost of an order does not depend on the composition of the order the joint and item dependent setup cost jis model allows for item dependent setup costs in addition to the joint setup costs we develop and analyze a class of so called progressive interval heuristics a progessive interval heuristic solves a js or jis problem over a progressively larger time interval always starting with period 1 but fixing the setup variables of a progressively larger number of periods at their optimal values in earlier iterations different variants in this class of heuristics allow for different degrees of flexibility in adjusting continuous variables determined in earlier iterations of the algorithm for the js model and the two basic implementations of the progressive interval heuristics we show under some mild parameter conditions that the heuristics can be designed to be e optimal for any desired value of e 0 with a running time that is polynomially bounded in the size of the problem they can also be designed to be simultaneously asymptotically optimal and polynomially bounded a numerical study covering both the js and jis models shows that a progressive interval heuristic generates close to optimal solutions with modest computational effort and that it can be effectively used to solve large scale problems

a branch and bound algorithm for a production scheduling problem in an assembly system under due date constraints abstract this paper focuses on a production scheduling problem in a tree structured assembly system operating on a make to order basis due dates are considered as constraints in the problem that is tardiness is not allowed the objective of the problem is to minimize holding costs of final product inventory as well as work in process inventory a mixed integer linear programming model is given and a branch and bound b b algorithm based on a lagrangian relaxation method is developed in the algorithm a subgradient method is used for obtaining good lower bounds of subproblems or partial solutions while a lagrangian heuristic is used for upper bounds results of computational experiments on randomly generated test problems showed that the b b algorithm found optimal solutions of problems of moderate sizes in a reasonable amount of computation time

a fast diffeomorphic image registration algorithm abstract this paper describes dartel which is an algorithm for diffeomorphic image registration it is implemented for both 2d and 3d image registration and has been formulated to include an option for estimating inverse consistent deformations nonlinear registration is considered as a local optimisation problem which is solved using a levenberg marquardt strategy the necessary matrix solutions are obtained in reasonable time using a multigrid method a constant eulerian velocity framework is used which allows a rapid scaling and squaring method to be used in the computations dartel has been applied to intersubject registration of 471 whole brain images and the resulting deformations were evaluated in terms of how well they encode the shape information necessary to separate male and female subjects and to predict the ages of the subjects

scheduling of design projects with uncertain number of iterations abstract a short product design cycle is critical to the success of companies in the era of time based competition the underlying design activities however are often interlinked and quite uncertain for example some activities may have to be iterated several times to meet the design criteria furthermore time critical projects suffer the risk of failure if they cannot meet established target dates generating good and robust schedules is thus critical especially under the concurrent engineering paradigm where the delay of a single task may have a domino effect on subsequent tasks and on other projects sharing designers and or resources this paper studies the scheduling of design projects with uncertain number of iterations while managing design risks a separable problem formulation that balances modeling accuracy and computation complexity is created with the goal to minimize project tardiness and risk penalties an optimization based methodology that combines lagrangian relaxation stochastic dynamic programming and ordinal optimization is developed numerical results supported by simulation demonstrate that near optimal solutions are obtained and uncertainties are effectively managed for problems of practical sizes

multi contrast large deformation diffeomorphic metric mapping for diffusion tensor imaging diffusion tensor imaging dti can reveal detailed white matter anatomy and has the potential to detect abnormalities in specific white matter structures such detection and quantification are however not straightforward the voxel based analysis after image normalization is one of the most widely used methods for quantitative image analyses to apply this approach to dti it is important to examine if structures in the white matter are well registered among subjects which would be highly dependent on employed algorithms for normalization in this paper we evaluate the accuracy of normalization of dti data using a highly elastic transformation algorithm called large deformation diffeomorphic metric mapping after simulation based validation of the algorithm dti data from normal subjects were used to measure the registration accuracy to examine the impact of morphological abnormalities on the accuracy the algorithm was also tested using data from alzheimer s disease ad patients with severe brain atrophy the accuracy level was measured by using manual landmark based white matter matching and surface based brain and ventricle matching as gold standard to improve the accuracy level cascading and multi contrast approaches were developed the accuracy level for the white matter was 1 88 0 55 and 2 19 0 84 mm for the measured locations in the controls and patients respectively

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

computation of diffusion function measures in q space using magnetic resonance hybrid diffusion imaging the distribution of water diffusion in biological tissues may be estimated by a 3 d fourier transform ft of diffusion weighted measurements in q space in this study methods for estimating diffusion spectrum measures the zero displacement probability the mean squared displacement and the orientation distribution function directly from the q space signals are described these methods were evaluated using both computer simulations and hybrid diffusion imaging hydi measurements on a human brain the hydi method obtains diffusion weighted measurements on concentric spheres in q space monte carlo computer simulations were performed to investigate effects of noise q space truncation and sampling interval on the measures this new direct computation approach reduces hydi data processing time and image artifacts arising from 3 d ft and regridding interpolation in addition it is less sensitive to the noise and q space truncation effects than conventional approach although this study focused on data using the hydi scheme this computation approach may be applied to other diffusion sampling schemes including cartesian diffusion spectrum imaging