projection results for vehicle routing a variety of integer programming formulations have been proposed for vehicle routing problems vrps including the so called two and three index formulations the set partitioning formulation and various formulations based on extra variables representing the flow of one or more commodities until now there has not been a systematic study of how these formulations relate to each other an exception is a paper of luis gouveia which shows that a one commodity flow formulation of gavish and graves yields by projection certain multistar inequalities in the two index space we give a survey of formulations for the capacitated vrp and then present various results of a similar flavour to those of gouveia in particular we show that the three index formulation augmented by certain families of valid inequalities gives the same lower bound as the two index formulation augmented by certain simpler families of valid inequalities the two commodity flow formulation of baldacci et al gives the same lower bound and the same multistar inequalities as the one commodity gavish and graves formulation a certain non standard multi commodity flow formulation with one commodity per customer implies by projection certain hypotour like inequalities in the two index space the set partitioning formulation implies by projection both multistar and hypotour like inequalities in the two index space we also briefly look at some other variants of the vrp such as the vrp with time windows and derive multistar like inequalities for them we also present polynomial time separation algorithms for some of the new inequalities

new developments in the primal dual column generation technique abstract the optimal solutions of the restricted master problems typically leads to an unstable behavior of the standard column generation technique and consequently originates an unnecessarily large number of iterations of the method to overcome this drawback variations of the standard approach use interior points of the dual feasible set instead of optimal solutions in this paper we focus on a variation known as the primal dual column generation technique which uses a primal dual interior point method to obtain well centered non optimal solutions of the restricted master problems we show that the method converges to an optimal solution of the master problem even though non optimal solutions are used in the course of the procedure also computational experiments are presented using linear relaxed reformulations of three classical integer programming problems the cutting stock problem the vehicle routing problem with time windows and the capacitated lot sizing problem with setup times the numerical results indicate that the appropriate use of a primal dual interior point method within the column generation technique contributes to a reduction of the number of iterations as well as the running times on average furthermore the results show that the larger the instance the better the relative performance of the primal dual column generation technique

optimal routing under capacity and distance restrictions this paper describes an integer linear programming algorithm for vehicle routing problems involving capacity and distance constraints the method uses constraint relaxation and a new class of subtour elimination constraints two versions of the algorithm are presented depending upon the nature of the distance matrix exact solutions are obtained for problems involving up to sixty cities

tolerance based branch and bound algorithms for the atsp abstract the selection of entries to be included excluded in branch and bound algorithms is usually done on the basis of cost values we consider the class of depth first search algorithms and we propose to use upper tolerances to guide the search for optimal solutions in spite of the fact that it needs time to calculate tolerances our computational experiments for asymmetric traveling salesman problems show that in most situations tolerance based algorithms outperform cost based algorithms the solution time reductions are mainly caused by the fact that the branching process becomes much more effective so that optimal solutions are found in an earlier stage of the branching process the use of tolerances also reveals why the widely used choice for branching on a smallest cycle in assignment solutions is on average the most effective one moreover it turns out that tolerance based dfs algorithms are better in solving difficult instances than the best first search algorithm from carpaneto et al carpaneto g dell amico m toth p 1995 exact solution of large scale asymmetric traveling salesman problems acm transactions on mathematical software 21 4 394 409

a standard test set for numerical approximations to the shallow water equations in spherical geometry a suite of seven test cases is proposed for the evaluation of numerical methods intended for the solution of the shallow water equations in spherical geometry the shallow water equations exhibit the major difficulties associated with the horizontal dynamical aspects of atmospheric modeling on the spherical earth these cases are designed for use in the evaluation of numerical methods proposed for climate modeling and to identify the potential trade offs which must always be made in numerical modeling before a proposed scheme is applied to a full baroclinic atmospheric model it must perform well on these problems in comparison with other currently accepted numerical methods the cases are presented in order of complexity they consist of advection across the poles steady state geostrophically balanced flow of both global and local scales forced nonlinear advection of an isolated low zonal flow impinging on an isolated mountain rossby haurwitz waves and observed atmospheric states one of the cases is also identified as a computer performance algorithm efficiency benchmark for assessing the performance of algorithms adapted to massively parallel computers

an addendum on sensitivity analysis of the optimal assignment we point out that sensitivity results for the linear assignment problem can be produced by a shortest path based approach in a straightforward manner and as efficient as finding an optimal solution

estimation of fractal signals from noisy measurements using wavelets the role of the wavelet transformation as a whitening filter for 1 f processes is exploited to address problems of parameter and signal estimations for 1 f processes embedded in white background noise robust computationally efficient and consistent iterative parameter estimation algorithms are derived based on the method of maximum likelihood and cramer rao bounds are obtained included among these algorithms are optimal fractal dimension estimators for noisy data algorithms for obtaining bayesian minimum mean square signal estimates are also derived together with an explicit formula for the resulting error these smoothing algorithms find application in signal enhancement and restoration the parameter estimation algorithms find application in signal enhancement and restoration the parameter estimation algorithms in addition to solving the spectrum estimation problem and to providing parameters for the smoothing process are useful in problems of signal detection and classification results from simulations are presented to demonstrated the viability of the algorithms

unsteady analytical solutions of the spherical shallow water equations a new class of unsteady analytical solutions of the spherical shallow water equations sswe is presented analytical solutions of the sswe are fundamental for the validation of barotropic atmospheric models to date only steady state analytical solutions are known from the literature the unsteady analytical solutions of the sswe are derived by applying the transformation method to the transition from a fixed cartesian to a rotating coordinate system fundamental examples of the new unsteady analytical solutions are presented for specific wind profiles with the presented unsteady analytical solutions one can provide a measure of the numerical convergence in the case of a temporally evolving system an application to the atmospheric model plasma shows the benefit of unsteady analytical solutions for the quantification of convergence properties

revisiting the rossby haurwitz wave test case with contour advection this paper re examines a basic test case used for spherical shallow water numerical models and underscores the need for accurate high resolution models of atmospheric and ocean dynamics the rossby haurwitz test case first proposed by williamson et al d l williamson j b drake j j hack r jakob p n swarztrauber a standard test set for numerical approximations to the shallow water equations on the sphere j comput phys 1992 221 224 has been examined using a wide variety of shallow water models in previous papers here two contour advective semi lagrangian casl models are considered and results are compared with previous test results we go further by modifying this test case in a simple way to initiate a rapid breakdown of the basic wave state this breakdown is accompanied by the formation of sharp potential vorticity gradients fronts placing far greater demands on the numerics than the original test case does we also go further by examining other dynamical fields besides the height and potential vorticity to assess how well the models deal with gravity waves such waves are sensitive to the presence or not of sharp potential vorticity gradients as well as to numerical parameter settings in particular large time steps convenient for semi lagrangian schemes can seriously affect gravity waves but can also have an adverse impact on the primary fields of height and velocity these problems are exacerbated by a poor resolution of potential vorticity gradients

approximate efficiency of a selection procedure for the number of regression variables we first show a simplification and a refinement of the result obtained by shibata 1976 it is an evaluation of the mean squared error of estimates of regression parameters when the number of regression variables is selected by a certain procedure the evaluation yields a good approximation to the efficiency of a selection procedure when the number of regression variables is small the approximate efficiency of fpe akaike 1970 aic akaike 1973 or cp mallows 1973 turns out to be not satisfactorily high although they are all asymptotically efficient in the sense of shibata 1981 based on such an approximation a procedure for choosing a out of the family fpe bhansali downham 1977 is suggested a further generalization is also developed consider a regression model y xji k l e where y yl yj is a vector of observations x is an n x k design matrix and f k fl1 fk 0 0 is a vector of regression parameters for simplicity we assume e el ej is a vector of independent normally distributed random variables with mean zero and variance a we call the above model model k in this paper we are concerned only with the problem of selecting the number k from a given range 1 k k under the model k we have a least squares estimate of the regression parameters 43 k h f1k which is a solution of x k x k k x k y here x k is an n x k submatrix consisting of the first k column vectors of the design matrix x hereafter f k is occasionally considered as the corresponding k dimensional vector with the undefined entries being zero suppose that observations yl yn come from a model ko 1 ko k with the regression parameter b such that f3k0 0 our main concern is the mean squared error

method of moving frames to solve the shallow water equations on arbitrary rotating curved surfaces a novel numerical scheme is proposed to solve the shallow water equations swes on arbitrary rotating curved surfaces based on the method of moving frames mmf in which the geometry is represented by orthonormal vectors the proposed scheme not only has the fewest dimensionality both in space and time but also does not require either of metric tensors composite meshes or the ambient space the mmf swe formulation is numerically discretized using the discontinuous galerkin method of arbitrary polynomial order pin space and an explicit runge kutta scheme in time the numerical model is validated against six standard tests on the sphere and the optimal order of convergence of p 1 is numerically demonstrated the mmf swe scheme is also demonstrated for its efficiency and stability on the general rotating surfaces such as ellipsoid irregular and non convex surfaces

convergence of discrete laplace beltrami operators over surfaces the convergence property of the discrete laplace beltrami operator is the foundationof convergence analysis of the numerical simulation process of some geometric partial differential equations which involve the operator the aim of this paper is to review several already used discrete laplace beltrami operators over triangulated surface and study numerically as well as theoretically their convergent behavior we show that none of them is convergent in general but two of them proposed by desbrun et al and meyer et al are convergent in a special case we point out that this special case is very important in the numerical simulation of geometric partial differential equations

a discontinuous galerkin method for the shallow water equations in spherical triangular coordinates a global model of the atmosphere is presented governed by the shallow water equations and discretized by a runge kutta discontinuous galerkin method on an unstructured triangular grid the shallow water equations on the sphere a two dimensional surface in r 3 are locally represented in terms of spherical triangular coordinates the appropriate local coordinate mappings on triangles on every triangular grid element this leads to a two dimensional representation of tangential momentum and therefore only two discrete momentum equations the discontinuous galerkin method consists of an integral formulation which requires both area elements and line element faces integrals here we use a rusanov numerical flux to resolve the discontinuous fluxes at the element faces a strong stability preserving third order runge kutta method is applied for the time discretization the polynomial space of order k on each curved triangle of the grid is characterized by a lagrange basis and requires high order quadature rules for the integration over elements and element faces for the presented method no mass matrix inversion is necessary except in a preprocessing step the validation of the atmospheric model has been done considering standard tests from williamson et al d l williamson j b drake j j hack r jakob p n swarztrauber a standard test set for numerical approximations to the shallow water equations in spherical geometry j comput phys 102 1992 211 224 unsteady analytical solutions of the nonlinear shallow water equations and a barotropic instability caused by an initial perturbation of a jet stream a convergence rate of o dx k 1 was observed in the model experiments furthermore a numerical experiment is presented for which the third order time integration method limits the model error thus the time step dt is restricted by both the cfl condition and accuracy demands conservation of mass was shown up to machine precision and energy conservation converges for both increasing grid resolution and increasing polynomial order k

a unified exact method for solving different classes of vehicle routing problems this paper presents a unified exact method for solving an extended model of the well known capacitated vehicle routing problem cvrp called the heterogenous vehicle routing problem hvrp where a mixed fleet of vehicles having different capacities routing and fixed costs is used to supply a set of customers the hvrp model considered in this paper contains as special cases the single depot cvrp all variants of the hvrp presented in the literature the site dependent vehicle routing problem sdvrp and the multi depot vehicle routing problem mdvrp this paper presents an exact algorithm for the hvrp based on the set partitioning formulation the exact algorithm uses three types of bounding procedures based on the lp relaxation and on the lagrangean relaxation of the mathematical formulation the bounding procedures allow to reduce the number of variables of the formulation so that the resulting problem can be solved by an integer linear programming solver extensive computational results over the main instances from the literature of the different variants of hvrps sdvrp and mdvrp show that the proposed lower bound is superior to the ones presented in the literature and that the exact algorithm can solve for the first time ever several test instances of all problem types considered

application of double fourier series to the shallow water equations on a sphere the spectral transform method using a double fourier series as orthogonal basis functions as in cheong 2000 j comput phys 157 327 is extended to the solution of the shallow water equations on a sphere a spectral filter which mimics the implicit diffusion process with the third order laplacian operator is applied to the spectral components of predicted variables to prevent the aliasing error or nonlinear instability for a predicted variable the spectral filter needs only about 76n2 operations with n being the zonal and meridional wavenumber truncation the use of the filter even at every time step does not deteriorate the computational efficiency of the double fourier series model which comes from the availability of ffts the filter requires an additional memory for only 6n2 elements so the total memory space of o n2 is sufficient in the present model along with the incorporation of the polar filter the semi implicit time stepping procedure contributes to a significant increase in the time step size a standard test set proposed by williamson et al 1992 j comput phys 102 211 is used to evaluate the errors associated with the new method for various resolutions it is shown that as a whole the accuracy of the method is comparable to that of spherical harmonics model shm though the present method provides more accurate time integration for some cases but does not for other cases a long time integration far beyond the period specified in the standard test set also illustrates almost the same accuracy as that given by the shm the relative efficiency of the method to the shm appears from the resolution of 256 128 transform grids and it becomes significant for resolutions higher than 512 256 the computational efficiency is expected to increase further with an improved fft algorithm the test results suggest that the new method could be extended to three dimensional numerical models used for weather prediction

time varying system identification and model validation using wavelets parametric identification of time varying tv systems is possible if each tv coefficient can be expanded onto a finite set of basis sequences the problem then becomes time invariant with respect to the parameters of the expansion the authors address the question of selecting this set of basis sequences they advocate the use of a wavelet basis because of its flexibility in capturing the signal s characteristics at different scales and discuss how to choose the optimal wavelet basis for a given system trajectory they also develop statistical tests to keep only the basis sequences that significantly contribute to the description of the system s time variation by formulating the problem as a regressor selection problem they apply an p test and an aic based approach for multiresolution analysis of tv systems the resulting algorithm can estimate tv ar or armax models and determine their orders they apply this algorithm to both synthetic and real speech data and compare it with the kalman filtering tv parameter estimator

cubic spline collocation method for the shallow water equations on the sphere spatial discretization schemes commonly used in global meteorological applications are currently limited to spectral methods or low order finite difference finite element methods the spectral transform method which yields high order approximations requires legendre transforms which have a computational complexity of o n3 where n is the number of subintervals in one dimension thus high order finite element methods may be a viable alternative to spectral methods in this study we present a new numerical method for solving the shallow water equations swe in spherical coordinates in this implementation the swe are discretized in time with the semi implicit leapfrog method and in space with the cubic spline collocation method on a skipped latitude longitude grid numerical results for the williamson et al swe test cases d l williamson j b blake j j hack r jakob and p n swarztrauber j comput phys 102 211 1992 are presented to demonstrate the stability and accuracy of the method results are also shown for an efficiency comparison between this method and a similar method in which spatial discretization is done on a uniform latitude longitude grid

new exact method for large asymmetric distance constrained vehicle routing problem in this paper we revise and modify an old branch and bound method for solving the asymmetric distance constrained vehicle routing problem suggested by laporte et al in 1987 our modification is based on reformulating distance constrained vehicle routing problem into a travelling salesman problem and on using assignment problem as a lower bounding procedure in addition our algorithm uses the best first strategy and new tolerance based branching rules since our method is fast but memory consuming it could stop before optimality is proven therefore we introduce the randomness in case of ties in choosing the node of the search tree if an optimal solution is not found we restart our procedure as far as we know the instances that we have solved exactly up to 1000 customers are much larger than the instances considered for other vehicle routing problem models from the recent literature so despite of its simplicity this proposed algorithm is capable of solving the largest instances ever solved in the literature moreover this approach is general and may be used for solving other types of vehicle routing problems

jackknife model averaging for quantile regressions abstract in this paper we consider model averaging for quantile regressions qr when all models under investigation are potentially misspecified and the number of parameters is diverging with the sample size to allow for the dependence between the error terms and regressors in the qr models we propose a jackknife model averaging jma estimator which selects the weights by minimizing a leave one out cross validation criterion function and demonstrate its asymptotic optimality in terms of minimizing the out of sample final prediction error we conduct simulations to demonstrate the finite sample performance of our estimator and compare it with other model selection and averaging methods we apply our jma method to forecast quantiles of excess stock returns and wages

a nodal triangle based spectral element method for the shallow water equations on the sphere a nodal triangle based spectral element se method for the shallow water equations on the sphere is presented the original se method uses quadrilateral elements and high order nodal lagrange polynomials constructed from a tensor product of the legendre gauss lobatto points in this work we construct the high order lagrange polynomials directly on the triangle using nodal sets obtained from the electrostatics principle j s hesthaven from electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex siam journal on numerical analysis 35 1998 655 676 and fekete points m a taylor b a wingate r e vincent an algorithm for computing fekete points in the triangle siam journal on numerical analysis 38 2000 1707 1720 these points have good approximation properties and far better lebesgue constants than any other nodal set derived for the triangle by employing triangular elements as the basic building blocks of the se method and the cartesian coordinate form of the equations we can use any grid imaginable including adaptive unstructured grids results for six test cases are presented to confirm the accuracy and stability of the method the results show that the triangle based se method yields the expected exponential convergence and that it can be more accurate than the quadrilateral based se method even while using 30 60 fewer grid points especially when adaptive grids are used to align the grid with the flow direction however at the moment the quadrilateral based se method is twice as fast as the triangle based se method because the latter does not yield a diagonal mass matrix

high order discontinuous galerkin schemes on general 2d manifolds applied to the shallow water equations an innovating approach is proposed to solve vectorial conservation laws on curved manifolds using the discontinuous galerkin method this new approach combines the advantages of the usual approaches described in the literature the vectorial fields are expressed in a unit non orthogonal local tangent basis derived from the polynomial mapping of curvilinear triangle elements while the convective flux functions are written is the usual 3d cartesian coordinate system the number of vectorial components is therefore minimum and the tangency constraint is naturally ensured while the method remains robust and general since not relying on a particular parametrization of the manifold the discontinuous galerkin method is particularly well suited for this approach since there is no continuity requirement between elements for the tangent basis definition the possible discontinuities of this basis are then taken into account in the riemann solver on inter element interfaces the approach is validated on the sphere using the shallow water equations for computing standard atmospheric benchmarks in particular the williamson test cases are used to analyze the impact of the geometry on the convergence rates for discretization error the propagation of gravity waves is eventually computed on non conventional irregular curved manifolds to illustrate the robustness and generality of the method

a general variable neighborhood search for the one commodity pickup and delivery travelling salesman problem abstract we present a variable neighborhood search approach for solving the one commodity pickup and delivery travelling salesman problem it is characterized by a set of customers such that each of the customers either supplies pickup customers or demands delivery customers a given amount of a single product and by a vehicle whose given capacity must not be exceeded that starts at the depot and must visit each customer only once the objective is to minimize the total length of the tour thus the considered problem includes checking the existence of a feasible travelling salesman s tour and designing the optimal travelling salesman s tour which are both np hard problems we adapt a collection of neighborhood structures k opt double bridge and insertion operators mainly used for solving the classical travelling salesman problem a binary indexed tree data structure is used which enables efficient feasibility checking and updating of solutions in these neighborhoods our extensive computational analysis shows that the proposed variable neighborhood search based heuristics outperforms the best known algorithms in terms of both the solution quality and computational efforts moreover we improve the best known solutions of all benchmark instances from the literature with 200 to 500 customers we are also able to solve instances with up to 1000 customers

a compatible and conservative spectral element method on unstructured grids we derive a formulation of the spectral element method which is compatible on very general unstructured three dimensional grids here compatible means that the method retains discrete analogs of several key properties of the divergence gradient and curl operators the divergence and gradient are anti adjoints the negative transpose of each other the curl is self adjoint and annihilates the gradient operator and the divergence annihilates the curl the adjoint relations hold globally and at the element level with the inclusion of a natural discrete element boundary flux term we then discretize the shallow water equations on the sphere using the cubed sphere grid and show that compatibility allows us to locally conserve mass energy and potential vorticity conservation is obtained without requiring the equations to be in conservation form the conservation is exact assuming exact time integration

sensitivity analysis of the optimal assignment abstract this paper concentrates on sensitivity analysis of the optimal solution for the assignment problem ap due to the high degeneracy of the ap traditional sensitivity analysis which determines the range in which the current optimal basis remains optimal is impractical thus changing the optimal basis does not ensure that the optimal assignment will be changed herein we investigate the properties of the ap and then propose several lemmas to determine two other types of sensitivity range the first type is used to determine the range in which the current optimal assignment remains optimal we further discuss what is the new optimal assignment when the changes surpass the range the second type of sensitivity range is to determine those values of assignment model parameters for which the rate of change of optimal value function remains constant an example is presented in order to demonstrate that the approaches are useful in practice