Extended Formulation Research Articles

Robust branch-cut-and-price for the Capacitated Minimum Spanning Tree problem over a large extended formulation

This paper presents a robust branch-cut-and-price algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to q-arbs, a structure that arises from a relaxation of the capacitated prize-collecting arborescence problem in order to make it solvable in pseudo-polynomial time. Traditional inequalities over the arc formulation, like Capacity Cuts, are also used. Moreover, a novel feature is introduced in such kind of algorithms: powerful new cuts expressed over a very large set of variables are added, without increasing the complexity of the pricing subproblem or the size of the LPs that are actually solved. Computational results on benchmark instances from the OR-Library show very significant improvements over previous algorithms. Several open instances could be solved to optimality.

Extension of cross section homogenization method for particle-dispersed media to layered particles

A cross section homogenization method for media containing randomly dispersed particles, which was originally developed by Shmakov et al., and then was improved by Yamamoto et al., is extended to layered particles, such as coated fuel particles of gas-cooled reactors. A new extended formulation is derived for concentric double-layered spherical particles. This formulation is successfully implemented into a continuous energy Monte Carlo code MCNP and is applied to some examples including a simplified gas-cooled reactor core. The extended method for double-layered spherical particles provides as accurate results as the previously developed method for mono-layered particles does. For the simple gas-cooled reactor core, the new method reproduces k eff within several hundreds of pcm as compared to the direct heterogeneous calculation in which double-layered spherical particles are explicitly distributed in the fuel compact. Whereas the previously developed method for mono-layered particles can be performed analytically, this new technique requires numerical integration, worsening the computation efficiency of the new method. However, this method allows one to perform Monte Carlo criticality calculations for double-layered particle-dispersed media with much less computing time than direct heterogeneous calculations.

Linear-programming extended formulations for the single-item lot-sizing problem with backlogging and constant capacity

Recently, several authors [8, 10] have argued for the use of extended formulations to tighten production planning models. In this work we present two linear-programming extended formulations of the constant-capacity lot-sizing problem with backlogging. The first one applies to the problem with a general cost function and has O(n3) variables and constraints. This improves on the more straightforward O(n4) Florian and Klein [2] type formulation. The second one applies when the costs satisfy the Wagner-Whitin property but it has O(n2) variables and O(n3) constraints. As a by-product, we positively answer an open question of Miller and Wolsey [4] about the tightness of an extended formulation for the continuous mixing set.

Lot-sizing with production and delivery time windows

We study two different lot-sizing problems with time windows that have been proposed recently. For the case of production time windows, in which each client specific order must be produced within a given time interval, we derive tight extended formulations for both the constant capacity and uncapacitated problems with Wagner-Whitin (non-speculative) costs. For the variant with nonspecific orders, known to be equivalent to the problem in which the time windows can be ordered by time, we also show equivalence to the basic lot-sizing problem with upper bounds on the stocks. Here we derive polynomial time dynamic programming algorithms and tight extended formulations for the uncapacitated and constant capacity problems with general costs.For the problem with delivery time windows, we use a similar approach to derive tight extended formulations for both the constant capacity and uncapacitated problems with Wagner-Whitin (non-speculative) costs.

Approximate extended formulations

Mixed integer programming (MIP) formulations are typically tightened through the use of a separation algorithm and the addition of violated cuts. Using extended formulations involving new variables is a possible alternative, but this often results in prohibitively large MIPs where even the linear programming relaxations are hard or impossible to solve. In this paper, we demonstrate how, in certain cases, it is possible and interesting to define ``approximate'' extended formulations. In all the examples considered, our description involves a single control parameter K. Large values of K result in strong but large formulations. In particular, when K takes its maximum value, the approximate formulation is identical to the complete extended formulation. Through this approximation parameter, the user has control over the tradeoff between the strength and the size of the formulation.

Lot-Sizing with Production and Delivery Time Windows

We study two different lot-sizing problems with time windows that have been proposed recently. For the case of production time windows, in which each client specific order must be produced within a given time interval, we derive tight extended formulations for both the constant capacity and uncapacitated problems with Wagner-Whitin (non-speculative) costs. For the variant with nonspecific orders, known to be equivalent to the problem in which the time windows can be ordered by time, we also show equivalence to the basic lot-sizing problem with upper bounds on the stocks. Here we derive polynomial time dynamic programming algorithms and tight extended formulations for the uncapacitated and constant capacity problems with general costs. For the problem with delivery time windows, we use a similar approach to derive tight extended formulations for both the constant capacity and uncapacitated problems with Wagner-Whitin (non-speculative) costs.

A NEW FORMULATION TOWARD UNIFYING THE VELOCITY ROLE IN COLLOCATED VARIABLE ARRANGEMENT

ABSTRACT One main challenge in numerical treatment of incompressible fluid flow problems is to suppress the decoupling of pressure and velocity fields. The challenge has prompted research toward suggesting and implementing various coupling strategies. In this work, a novel strategy which suitably couples pressure and velocity in a collocated grid arrangement is presented. The current strategy develops a unique cell-face velocity expression which provides infinite cell-face velocity magnitudes in the algorithm. A smoothing factor is incorporated in the cell-face velocity expression in order to produce a wide range of velocity magnitudes. The smoothing factor provides a smooth transition from an unreal zigzag pressure field to an acceptable physical distribution. The extended formulation is then examined in a domain with a source and a sink which represent either velocity or pressure discontinuities in the domain. The discontinuities may in turn lead to nonphysical solution if the pressure–velocity coupling is weak or inappropriate. The current investigation shows that the new extended formulation provides a robust approach to cure the decoupling problem.

A solution approach of production planning problems based on compact formulations for single-item lot-sizing models

We survey the main results presented in the author’s PhD Thesis presented in June 2003 at the Universite catholique de Louvain and supervised by Y. Pochet and L.A. Wolsey. The dissertation is written in English and is available from the author. In the first part of the thesis, we investigate the complexity and the polyhedral structure of various extensions of the uncapacitated single-item lot-sizing problem (Barany et al. 1984). In particular, we study models involving fixed charges on stocks, constant capacity and backlogging, and lower bounds on production. We describe algorithms, extended formulations, (facet-defining) valid inequalities and separation algorithms. Emphasis is placed on compact (i.e. of polynomial size) exact extended formulations. In a second part, we show how such extended reformulations for single-item problems can help to improve the solution of much more general production planning problems.

Strong formulations for mixed integer programs: valid inequalities and extended formulations

We examine progress over the last fifteen years in finding strong valid inequalities and tight extended formulations for simple mixed integer sets lying both on the ``easy'' and ``hard'' sides of the complexity frontier. Most progress has been made in studying sets arising from knapsack and single node flow sets, and a variety of sets motivated by different lot-sizing models. We conclude by citing briefly some of the more intriguing new avenues of research.

The uncapacitated lot-sizing problem with sales and safety stocks

We examine a variant of the uncapacitated lot-sizing model of Wagner-Whitin involving sales instead of fixed demands, and lower bounds on stocks. Two extended formulations are presented, as well as a dynamic programming algorithm and a complete description of the convex hull of solutions. When the lower bounds on stocks are non-decreasing over time, it is possible to describe an extended formulation for the problem and a combinatorial separation algorithm for the convex hull of solutions. Finally when the lower bounds on stocks are constant, a simpler polyhedral description is obtained for the case of Wagner-Whitin costs.

Anticommutative Electric and Magnetic Charges, and the Monopole Question

The monopole question is treated anew in the light of a recent, strictly covariant,extended formulation of fermion quantum field theory naturally including alsoa pseudoscalar variety of conserved charges. The essential novelty lies in theresulting quantum property of anticommutivity between scalar and pseudoscalarcharge varieties, which should in particular apply to electric and magnetic charges.As an immediate outcome, there should no longer be any (Dirac-like) quantizationcondition relating these charges and binding the magnetic elementary charge tohave a very great strength. A generalized Lagrangian approach to the monopoleproblem is made truly viable, leading to two independent local gauge couplingswhich are separately generated by the electric and magnetic elementary chargesand are not allowed to interfere. This would prevent electric and magneticmonopoles from mutually interacting and would particularly account for the“absence” of magnetic sources in ordinary electromagnetism. Within such aframework, an electric charge eigenstate with a nonzero eigenvalue is bound tohave a null magnetic charge expectation value, and the magnetic dipole momentof an electrically charged point fermion may actually be seen as resulting fromthe additional internal presence of a single magnetic charge subjected to a maximaluncertainty in sign. An easy estimate makes it allowable to assign to this chargea strength just equal to that of the partner electric charge. Such a conjecture leadsto a “dual” model of a charged point fermion where the “electric” and “magnetic”roles can well be interchanged with no observable effects. In the associatedformalism, duality symmetry is already included without the need to appeal toany “missing” electromagnetic phenomenology to be discovered.

Heat Transfer Through an Extended Surface Containing He II

A semi-analytic solution for the heat transfer process between a He II pressurized bath and a saturated tube-type heat exchanger is presented. The problem is modeled with an extended surface heat transfer formulation analogous to that in conventional conduction. The process is governed by Kapitza conductance and counterflow within the bulk fluid in the tube. The resulting nonlinear differential equation may be integrated for the special case of constant properties, yielding a simple solution applicable to design and analysis of practical heat exchangers.

Source sink flows with capacity installation in batches

We consider the problem of sending flow from a source to a destination where there are flow costs on each arc and fixed costs toward the purchase of capacity. Capacity can be purchased in batches of C units on each arc. We show the problem to be NP-hard in general. If d is the quantity to be shipped from the source to the destination, we give an algorithm that solves the problem in time polynomial in the size of the graph but exponential in [ d C ]. Thus, for bounded values of [ d C ]the problem can be solved in polynomial time. This is useful since a simple heuristic gives a very good approximation of the optimal solution for large values of [ d C ]. We also show a similar result to hold for the case when there are no flow costs but capacity can be purchased either in batches of 1 unit or C units. The results characterizing optimal solutions with a minimum number of free arcs are used to obtain extended formulations in each of the two cases. The LP-relaxations of the extended formulations are shown to be stronger than the natural formulations considered by earlier authors, even with a family of strong valid inequalities added.

Extended impedance control of redundant manipulators based on weighted decomposition of joint space

In this work, impedance control approach based on an extended task space formulation is addressed to control the kinematically redundant manipulators. By defining a weighted inner product in joint space, a minimal parameterization of the null space is achieved, and we can visualize the null space motion explicitly. Moreover, it is shown that careful choice of the weighting matrix gives physically consistent and inertially decoupled dynamics. By augmenting this minimal null motion parameter with a forward kinematic relation, a new extended task space formulation can be obtained. Based on this formulation, we propose two control methods, a kinematically decomposed impedance controller and an inertially decoupled impedance controller, to control the motion of the end-effector as well as the internal motion expanding the conventional impedance control. We also show the relationship with the previous dynamic controllers of a redundant manipulator. Some numerical simulations are given to demonstrate the performance of the proposed control methods. © 1998 John Wiley & Sons, Inc.

Scattering from a cavity backed microstrip antenna on a bi-anisotropic substrate with a finite ground plane by a hybrid finite element method

A new extended finite element formulation for finding radiation patterns, radar cross section (RCS) and input impedance of planar integrated circuits on a very general bi-anisotropic grounded slab with finite size is presented. The approach is based on a hybrid finite element method (FEM) wherein the electromagnetic fields in the FEM region are coupled to the fields outside by a surface integral equation. Because of the local nature of the FEM, this formalism allows inclusion of the inhomogeneities in the problem domain. Numerical results are compared and agree well with existing published data for various conventional configurations.

Analytical solution of perturbed circular motion: application to satellite geodesy

Starting from the analytical theory of perturbed␣circular motions presented in Celestial Mechanics (Bois 1994) and from specific extended formulations of the perturbations in a uniformly rotating plane of constant inclination, this paper presents an extended formulation of the solution. The actual gain made through this extension is the establishment of a first-order predictive theory written in spherical coordinates and thus free of singularities, whose perturbations are directly expressed in the local orbital frame generally used in satellite geodesy. This new formulation improves the generality, the precision and the field of applications of the theory. It is particularly devoted to the analysis of satellite position perturbations for satellites in low eccentricity orbits usually used for many Earth observation applications. An application to the TOPEX/Poseidon (T/P) orbit is performed. In particular, contour maps are provided which show the geographical location of orbit differences coming from geopotential coefficient differences of two recent gravity field models. Comparison of predicted radial and along-track orbit differences with respect to numerical results provided by the French group (CNES, in Toulouse) in charge of the T/P orbit are convincing.

Extended formulations for the A-cut problem

LetG=(V, E) be an undirected graph andA⊆V. We consider the problem of finding a minimum cost set of edges whose deletion separates every pair of nodes inA. We consider two extended formulations using both node and edge variables. An edge variable formulation has previously been considered for this problem (Chopra and Rao (1991), Cunningham (1991)). We show that the LP-relaxations of the extended formulations are stronger than the LP-relaxation of the edge variable formulation (even with an extra class of valid inequalities added). This is interesting because, while the LP-relaxations of the extended formulations can be solved in polynomial time, the LP-relaxation of the edge variable formulation cannot. We also give a class of valid inequalities for one of the extended formulations. Computational results using the extended formulations are performed.

Analytical solution of the perturbed circular motion: an extended formulation for various perturbations

An extended formulation of the solution is presented starting from the analytical theory of perturbed circular motions developed by Bois ( Celestial Mech. 58, 125–138, 1994) and from specific extended formulations of the perturbations. The field of applications of the theory is then increased. A treatment of several perturbations is presented such as an external potential, a non-gravitational force, both added to an extended central potential. The expanded solution permits to take into account immediately and simultaneously different periodic perturbations. It is then directly applicable to problems of various physical nature. In the present numerical applications devoted to artificial satellites with zero inclination, physical descriptions are given in each case. Over few orbits, the solutions are particularly accurate and the comparisons of the results with respect to a numerical integration are convincing. Moreover, the long period evolution over about a thousand orbits is illustrated stemming both from the solution and the numerical integration.

Dynamics of multicomponent polymer mixtures via the random phase approximation including hydrodynamic interactions

A new and extended formulation of the random phase approximation (RPA) in the study of statics and dynamics of multicomponent polymer mixtures is presented. The new formulation simplifies the implementation of the RPA in both compressible and incompressible mixtures and allows the inclusion of hydrodynamic interaction in the dynamics of polymer melts in the RPA. The dynamics of copolymer melts with hydrodynamic interaction is studied in detail as an illustration of the extended formulation, and the variation of the first cumulant as a function of the wavenumber and interaction parameter is obtained. In this paper we present a generalization of the above elimination procedure by allowing the monomers of the matrix component to interact or to be connected with the monomers in other components, so that the assumption xjo(q,s) = 0, for j = 1, ..., n is no longer needed. This generalization not only simplifies the implementation of the RPA in incompressible multicomponent mixtures in which none of the components consists of homopolymers, such as in melts of copolymers, stars, etc., but also it allows more flexibility in the choice of the bare system in the implementation of the RPA, even in the case of ho-

Development of a Sensitivity Analysis Method in Prostbuckling Problems.

An analysis method to evaluate the sensitivities in postbuckling problems is developed. In recent years, sensitivity analysis using the perturbation method has been applied to nonlinear static/dynamic problems. In postbuckling analysis, however, the load values at limit or bifurcation points are also unknown variables of which the gradients cannot be evaluated by the conventional sensitivity analysis. The new extended formulation can calculate the gradients of the load values as well as the displacement gradients under a constant 'arc-length' condition, which is generally used as a path-following method to find the equilibrium state beyond the limit or bifurcation points. A couple of buckling analyses of sheli structures are demonstrated to examine the validity of this method, and satisfactory results are obtained.