Adapting the interior point method for the solution of linear programs on high performance computers

In this paper we describe a unified algorithmic framework for the interior point method (IPM) of solving Linear Programs (LPs) which allows us to adapt it over a range of high performance computer architectures. We set out the reasons as to why IPM makes better use of high performance computer architecture than the sparse simplex method. In the inner iteration of the IPM a search direction is computed using Newton or higher order methods. Computationally this involves solving a sparse symmetric positive definite (SSPD) system of equations. The choice of direct and indirect methods for the solution of this system and the design of data structures to take advantage of coarse grain parallel and massively parallel computer architectures are considered in detail. Finally, we present experimental results of solving NETLIB test problems on examples of these architectures and put forward arguments as to why integration of the system within sparse simplex is beneficial

Developments in linear and integer programming

In this review we describe recent developments in linear and integer (linear) programming. For over 50 years Operational Research practitioners have made use of linear optimisation models to aid decision making and over this period the size of problems that can be solved has increased dramatically, the time required to solve problems has decreased substantially and the flexibility of modelling and solving systems has increased steadily. Large models are no longer confined to large computers, and the flexibility of optimisation systems embedded in other decision support tools has made on-line decision making using linear programming a reality (and using integer programming a possibility). The review focuses on recent developments in algorithms, software and applications and investigates some connections between linear optimisation and other technologies

Combining Column Generation and Lagrangian Relaxation

Although the possibility to combine column generation and Lagrangian relaxation has been known for quite some time, it has only recently been exploited in algorithms. In this paper, we discuss ways of combining these techniques. We focus on solving the LP relaxation of the Dantzig-Wolfe master problem. In a first approach we apply Lagrangian relaxation directly to this extended formulation, i.e. no simplex method is used. In a second one, we use Lagrangian relaxation to generate new columns, that is Lagrangian relaxation is applied to the compact for-mulation. We will illustrate the ideas behind these algorithms with an application in Lot-sizing. To show the wide applicability of these techniques, we also discuss applications in integrated vehicle and crew scheduling, plant location and cutting stock problems.column generation;Lagrangean relaxation;cutting stock problem;lotsizing;vehicle and crew scheduling

A computational comparison of two simplicial decomposition approaches for the separable traffic assignment problems : RSDTA and RSDVI

Draft pel 4th Meeting del Euro Working Group on Transportation (Newcastle 9-11 setembre de 1.996)The class of simplicial decomposition methods has shown to constitute efficient tools for the solution of the variational inequality formulation of the general traffic assignment problem. The paper presents a particular implementation of such an algorithm, called RSDVI, and a restricted simplicial decomposition algorithm, developed adhoc for diagonal, separable, problems named RSDTA. Both computer codes are compared for large scale separable traffic assignment problems. Some meaningful figures are shown for general problems with several levels of asymmetry.Preprin

Advances in design and implementation of optimization software

Developing optimization software that is capable of solving large and complex real-life problems is a huge effort. It is based on a deep knowledge of four areas: theory of optimization algorithms, relevant results of computer science, principles of software engineering, and computer technology. The paper highlights the diverse requirements of optimization software and introduces the ingredients needed to fulfill them. After a review of the hardware/software environment it gives a survey of computationally successful techniques for continuous optimization. It also outlines the perspective offered by parallel computing, and stresses the importance of optimization modeling systems. The inclusion of many references is intended to both give due credit to results in the field of optimization software and help readers obtain more detailed information on issues of interest

PyDEC: Software and Algorithms for Discretization of Exterior Calculus

This paper describes the algorithms, features and implementation of PyDEC, a Python library for computations related to the discretization of exterior calculus. PyDEC facilitates inquiry into both physical problems on manifolds as well as purely topological problems on abstract complexes. We describe efficient algorithms for constructing the operators and objects that arise in discrete exterior calculus, lowest order finite element exterior calculus and in related topological problems. Our algorithms are formulated in terms of high-level matrix operations which extend to arbitrary dimension. As a result, our implementations map well to the facilities of numerical libraries such as NumPy and SciPy. The availability of such libraries makes Python suitable for prototyping numerical methods. We demonstrate how PyDEC is used to solve physical and topological problems through several concise examples.Comment: Revised as per referee reports. Added information on scalability, removed redundant text, emphasized the role of matrix based algorithms, shortened length of pape

Reformulation and decomposition of integer programs

In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm