Integer-friendly formulations for the r-separation problem

In this paper, we consider different formulations for the r-separation problem, where the objective is to choose as as many points as possible from a given set of points subject to the constraint that no two selected points can be closer than r units to one another. Our goal is to devise a mathematical programming formulation with an LP-relaxation which yields integer solutions with great frequency. We consider six different formulations of the r-separation problem. We show that the LP-relaxations of the most obvious formulations will yield fractional results in all instances of the problem if an optimal solution contains fewer than half of the given points. To build computationally effective formulations for the r-separation problem, we write dense constraints with unit right-hand-sides. The LP formulation that performs the best in our computational tests almost always finds 0–1 solutions to the problem.     

New formulations for the uncapacitated multiple allocation hub location problem

In this paper we review the integer linear formulations of the uncapacitated multiple allocation hub location problem, we study the scope of validity of these formulations and give new ones that generalize the older formulations. Our formulations allow one or two visits to hubs and include a more general cost structure that needs not satisfy the triangle inequality. We prove that the constraints defined by cliques of a related (intersection) graph are tighter constraints than the classical ones. We also discuss a pre-processing of the problem, which is very useful for reducing its size. Finally, we check the strength of the new formulations and compare them with others in the literature by solving instances of two commonly used data sets: the CAB (Civil Aeronautics Board) and AP (Australian Post), and also randomly generated instances. Our computational results clearly show that our formulations outperform all others previously used for small and medium problems.     

On the equivalence of strong formulations for capacitated multi-level lot sizing problems with setup times

Several mixed integer programming formulations have been proposed for modeling capacitated multi-level lot sizing problems with setup times. These formulations include the so-called facility location formulation, the shortest route formulation, and the inventory and lot sizing formulation with (?, S) inequalities. In this paper, we demonstrate the equivalence of these formulations when the integrality requirement is relaxed for any subset of binary setup decision variables. This equivalence has significant implications for decomposition-based methods since same optimal solution values are obtained no matter which formulation is used. In particular, we discuss the relax-and-fix method, a decomposition-based heuristic used for the efficient solution of hard lot sizing problems. Computational tests allow us to compare the effectiveness of different formulations using benchmark problems. The choice of formulation directly affects the required computational effort, and our results therefore provide guidelines on choosing an effective formulation during the development of heuristic-based solution procedures.     

On formulations of the stochastic uncapacitated lot-sizing problem

We consider two formulations of a stochastic uncapacitated lot-sizing problem. We show that by adding (?,S) inequalities to the one with the smaller number of variables, both formulations give the same LP bound. Then we show that for two-period problems, adding another class of inequalities gives the convex hull of integral solutions.     

An analytical comparison of the LP relaxations of integer models for the k-club problem

Given an undirected graph G = (V, E), a k-club is a subset of nodes that induces a subgraph with diameter at most k. The k-club problem is to find a maximum cardinality k-club. In this study, we use a linear programming relaxation standpoint to compare integer formulations for the k-club problem. The comparisons involve formulations known from the literature and new formulations, built in different variable spaces. For the case k = 3, we propose two enhanced compact formulations. From the LP relaxation standpoint these formulations dominate all other compact formulations in the literature and are equivalent to a formulation with a non-polynomial number of constraints. Also for k = 3, we compare the relative strength of LP relaxations for all formulations examined in the study (new and known from the literature). Based on insights obtained from the comparative study, we devise a strengthened version of a recursive compact formulation in the literature for the k-club problem (k > 1) and show how to modify one of the new formulations for the case k = 3 in order to accommodate additional constraints recently proposed in the literature.     

Mixed-finite element and finite volume discretization for heavy brine simulations in groundwater

Recently, a new theory of high-concentration brine transport in groundwater has been developed. This approach is based on two nonlinear mass conservation equations, one for the fluid (flow equation) and one for the salt (transport equation), both having nonlinear diffusion terms. In this paper, we present and analyze a numerical technique for the solution of such a model. The approach is based on the mixed hybrid finite element method for the discretization of the diffusion terms in both the flow and transport equations, and a high-resolution TVD finite volume scheme for the convective term. This latter technique is coupled to the discretized diffusive flux by means of a time-splitting approach. A commonly used benchmark test (Elder problem) is used to verify the robustness and nonoscillatory behavior of the proposed scheme and to test the validity of two different formulations, one based on using pressure head ψ and concentration c as dependent variables, and one using pressure p and mass fraction ω as dependent variables. It is found that the latter formulation gives more accurate and reliable results, in particular, at large times. The numerical model is then compared against a semi-analytical solution and the results of a laboratory test. These tests are used to verify numerically the performance and robustness of the proposed numerical scheme when high-concentration gradients (i.e., the double nonlinearity) are present.     

Location problems

Recent developments for several location problems are surveyed. These include: graph theoretic and combinatorial formulations of the simple plant location problem, the NP-hardness of some p-center problems, worst-case bounds for several polynomial-time heuristics for some p-center problems, and a general solution to a class of one facility network problems with convex cost functions.     

Minimal spanning trees with a constraint on the number of leaves

In this paper we discuss minimal spanning trees with a constraint on the number of leaves. Tree topologies appear when designing centralized terminal networks. The constraint on the number of leaves arises because the software and hardware associated to each terminal differs accordingly with its position in the tree. Usually, the software and hardware associated to a “degree-1” terminal is cheaper than the software and hardware used in the remaining terminals because for any intermediate terminal j one needs to check if the arrival message is destined to that node or to any other node located after node j. As a consequence, that particular terminal needs software and hardware for message routing. On the other hand, such equipment is not needed in “degree-1” terminals. Assuming that the hardware and software for message routing in the nodes is already available, the above discussion motivates a constraint stating that a tree solution has to contain exactly a certain number of “degree-1” terminals. We present two different formulations for this problem and some lower bounding schemes derived from them. We discuss a simple local-exchange heuristic and present computational results taken from a set of complete graphs with up to 40 nodes. Integer Linear Programming formulations for related problems are also discussed at the end.     

Lower Bounds to Eigenvalues for Vibrations and Buckling of Structures by Analytical Applications of a-priori Known Qualitative Information

Two methods to obtain lower bounds to eigenvalues are presented for cases which have equivalent minimum variational formulations. One method is an extension and elaboration of a theorem presented by the author in 1972, which affected the transfer of a weight function from one location to another over the physical system considered. The extension relies on information known a-priori about the exact solution of the problem, although the exact solution is not obtained. The other method is akin to the Rayleigh–Ritz method but yields lower bounds. The two methods are applied to various physical examples of vibrations and of buckling with rather good results. The application to other examples is direct and may be performed in a way quite similar to those examples shown.     

The newsstand problem: A capacitated multiple-product single-period inventory problem

Practically all of the many earlier papers on the newsboy problem consider a single newsboy product with no capacity constraint. This paper presents formulations and solution procedures for handling multiple newsboy-type products under two cases: (A) one resource-capacity constraint; and (B) multiple resource constraints. For case A, we present a necessary extension to the classical Hadley-Whitin result for handling general demand distributions. For case B we present a solution procedure that can efficiently handle the common situation where a very large number of products are involved.     

New formulations of the Hop-Constrained Minimum Spanning Tree problem via Miller-Tucker-Zemlin constraints

Given an undirected network with positive edge costs and a natural number p, the Hop-Constrained Minimum Spanning Tree problem (HMST) is the problem of finding a spanning tree with minimum total cost such that each path starting from a specified root node has no more than p hops (edges). In this paper, we develop new formulations for HMST. The formulations are based on Miller-Tucker-Zemlin (MTZ) subtour elimination constraints, MTZ-based liftings in the literature offered for HMST, and a new set of topology-enforcing constraints. We also compare the proposed models with the MTZ-based models in the literature with respect to linear programming relaxation bounds and solution times. The results indicate that the new models give considerably better bounds and solution times than their counterparts in the literature and that the new set of constraints is competitive with liftings to MTZ constraints, some of which are based on well-known, strong liftings of Desrochers and Laporte (1991).     

Relaxation approximation to bed-load sediment transport

In this work we propose and apply a numerical method based on finite volume relaxation approximation for computing the bed-load sediment transport in shallow water flows, in one and two space dimensions. The water flow is modeled by the well-known nonlinear shallow water equations which are coupled with a bed updating equation. Using a relaxation approximation, the nonlinear set of equations (and for two different formulations) is transformed to a semilinear diagonalizable problem with linear characteristic variables. A second order MUSCL-TVD method is used for the advection stage while an implicit–explicit Runge–Kutta scheme solves the relaxation stage. The main advantages of this approach are that neither Riemann problem solvers nor nonlinear iterations are required during the solution process. For the two different formulations, the applicability and effectiveness of the presented scheme is verified by comparing numerical results obtained for several benchmark test problems.     

Trois formulations d'un modèle de plasma quasi-neutre avec courant non nul

In this Note, we propose three formulations of a model describing a quasi-neutral plasma with non-vanishing current. In order to study and compare the numerical efficiency of each formulation, two test-problems are implemented in one dimension. The first is a periodic perturbation of a uniform stationary plasma. The second is a case of plasma expansion in vacuum between two electrodes. To cite this article: P. Crispel et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Compact formulations of the Steiner Traveling Salesman Problem and related problems

The Steiner Traveling Salesman Problem (STSP) is a variant of the TSP that is particularly suitable when routing on real-life road networks. The standard integer programming formulations of both the TSP and STSP have an exponential number of constraints. On the other hand, several compact formulations of the TSP, i.e., formulations of polynomial size, are known. In this paper, we adapt some of them to the STSP, and compare them both theoretically and computationally. It turns out that, just by putting the best of the formulations into the CPLEX branch-and-bound solver, one can solve instances with over 200 nodes. We also briefly discuss the adaptation of our formulations to some related problems.     

Hop constrained Steiner trees with multiple root nodes

We consider a network design problem that generalizes the hop and diameter constrained Steiner tree problem as follows: Given an edge-weighted undirected graph with two disjoint subsets representing roots and terminals, find a minimum-weight subtree that spans all the roots and terminals so that the number of hops between each relevant node and an arbitrary root does not exceed a given hop limit H. The set of relevant nodes may be equal to the set of terminals, or to the union of terminals and root nodes. This article proposes integer linear programming models utilizing one layered graph for each root node. Different possibilities to relate solutions on each of the layered graphs as well as additional strengthening inequalities are then discussed. Furthermore, theoretical comparisons between these models and to previously proposed flow- and path-based formulations are given. To solve the problem to optimality, we implement branch-and-cut algorithms for the layered graph formulations. Our computational study shows their clear advantages over previously existing approaches.     

Alternative formulations of a combined trip generation,trip distribution,modal split,and trip assignment model

The traditional four-step model has been widely used in travel demand forecasting by considering trip generation, trip distribution, modal split and traffic assignment sequentially in a fixed order. However, this sequential approach suffers from the inconsistency among the level-of-service and flow values in each step of the procedure. In the last two decades, this problem has been addressed by many researchers who have sought to develop combined (or integrated) models that can consider travelers’ choice on different stages simultaneously and give consistent results. In this paper, alternative formulations, including mathematical programming (MP) formulation and variational inequality (VI) formulations, are provided for a combined travel demand model that integrates trip generation, trip distribution, modal split, and traffic assignment using the random utility theory framework. Thus, the proposed alternative formulations not only allow a systematic and consistent treatment of travel choice over different dimensions but also have behavioral richness. Qualitative properties of the formulations are also given to ensure the existence and uniqueness of the solution. Particularly, the model is analyzed for a special but useful case where the probabilistic travel choices are assumed to be a hierarchical logit model. Furthermore, a self-adaptive Goldstein–Levitin–Polyak (GLP) projection algorithm is adopted for solving this special case.     

On relaxations of the max k-cut problem formulations

A tight continuous relaxation is a crucial factor in solving mixed integer formulations of many NP-hard combinatorial optimization problems. The (weighted) max k-cut problem is a fundamental combinatorial optimization problem with multiple notorious mixed integer optimization formulations. In this paper, we explore four existing mixed integer optimization formulations of the max k-cut problem. Specifically, we show that the continuous relaxation of a binary quadratic optimization formulation of the problem is: (i) stronger than the continuous relaxation of two mixed integer linear optimization formulations and (ii) at least as strong as the continuous relaxation of a mixed integer semidefinite optimization formulation. We also conduct a set of experiments on multiple sets of instances of the max k-cut problem using state-of-the-art solvers that empirically confirm the theoretical results in item (i). Furthermore, these numerical results illustrate the advances in the efficiency of global non-convex quadratic optimization solvers and more general mixed integer nonlinear optimization solvers. As a result, these solvers provide a promising option to solve combinatorial optimization problems. Our codes and data are available on GitHub.     

SVM classification for imbalanced data sets using a multiobjective optimization framework

Classification of imbalanced data sets in which negative instances outnumber the positive instances is a significant challenge. These data sets are commonly encountered in real-life problems. However, performance of well-known classifiers is limited in such cases. Various solution approaches have been proposed for the class imbalance problem using either data-level or algorithm-level modifications. Support Vector Machines (SVMs) that have a solid theoretical background also encounter a dramatic decrease in performance when the data distribution is imbalanced. In this study, we propose an L ₁-norm SVM approach that is based on a three objective optimization problem so as to incorporate into the formulation the error sums for the two classes independently. Motivated by the inherent multi objective nature of the SVMs, the solution approach utilizes a reduction into two criteria formulations and investigates the efficient frontier systematically. The results indicate that a comprehensive treatment of distinct positive and negative error levels may lead to performance improvements that have varying degrees of increased computational effort.     

Lower bounds for nonlinear assignment problems using many body interactions

This paper concerns lower bounding techniques for the general α-adic assignment problem. The nonlinear objective function is linearized by the introduction of additional variables and constraints, thus yielding a mixed integer linear programming formulation of the problem. The concept of many body interactions is introduced to strengthen this formulation and incorporated in a modified formulation obtained by lifting the original representation to a higher dimensional space. This process involves two steps — (i) addition of new variables and constraints and (ii) incorporation of the new variables in the objective function. If this lifting process is repeated β times on an α-adic assignment problem along with the incorporation of higher order interactions, it results in the mixed-integer formulation of an equivalent (α + β)-adic assignment problem. The incorporation of many body interactions in the higher dimensional formulation improves its degeneracy properties and is also critical to the derivation of decomposition methods for the solution of these large scale mathematical programs in the higher dimensional space. It is shown that a lower bound to the optimal solution of the corresponding linear programming relaxation can be obtained by dualizing a subset of constraints in this formulation and solving O(N^2(α+β−1)) linear assignment problems, whose coefficients depend on the dual values. Moreover, it is proved that the optimal solution to the LP relaxation is obtained if we use the optimal duals for the solution of the linear assignment problems. This concept of many body interactions could be applied in designing algorithms for the solution of formulations obtained by lifting general MILP's. We illustrate all these concepts on the quadratic assignment problems With these decomposition bounds, we have found the provably optimal solutions of two unsolved QAP's of size 32 and have also improved upon existing lower bounds for other QAP's.     

Numerical stability of the BEM for advection‐diffusion problems

A boundary element method (BEM) approach has been developed to solve the time‐dependent 1D advection‐diffusion equation. The 1D solution is part of a 3D numerical scheme for solving advection‐diffusion (AD) problems in fractured porous media. The full 3D scheme includes a 3D solution for the porous matrix, which is coupled with a 2D solution for fractures and a 1D solution for fracture intersections. As the hydraulic conductivity of the fracture intersections is usually higher than the hydraulic conductivity of the fractures and by at least one order of magnitude higher than the hydraulic conductivity of the porous matrix, the fastest flow and solute transport occurs in the fracture intersections. Therefore it is important to have an accurate and stable 1D solution of the transient AD problems. This article presents two different 1D BEM formulations for solution of the AD problems. The particular advantage of these formulations is that they provide one of the most straightforward and simplest ways to couple multiple intersecting 2D Boundary Element problems discretized with linear discontinuous elements. Both formulations are tested and compared for accuracy, stability, and consistency. The analysis helps to select the more suitable formulations according to the properties of the problem under consideration. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004