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

Integrated risk/cost planning models for the US Air Traffic system

A prototype network planning model for the U.S. Air Traffic control system is described. The model encompasses the dual objectives of managing collision risks and transportation costs where traffic flows can be related to these objectives. The underlying structure is a network graph with nonseparable convex costs; the model is solved efficiently by capitalizing on its intrinsic characteristics. Two specialized algorithms for solving the resulting problems are described: (1) truncated Newton, and (2) simplicial decomposition. The feasibility of the approach is demonstrated using data collected from a control center in the Midwest. Computational results with different computer systems are presented, including a vector supercomputer (CRAY-XMP). The risk/cost model has two primary uses: (1) as a strategic planning tool using aggregate flight information, and (2) as an integrated operational system for forecasting congestion and monitoring (controlling) flow throughout the U.S. In the latter case, access to a supercomputer is required due to the model's enormous size

An algorithm for determining copositive matrices

In this paper, we present an algorithm of simple exponential growth called COPOMATRIX for determining the copositivity of a real symmetric matrix. The core of this algorithm is a decomposition theorem, which is used to deal with simplicial subdivision of $\hat{T}^{-}=\{y\in \Delta_{m}| \beta^Ty\leq 0\}$ on the standard simplex $\Delta_m$, where each component of the vector $\beta$ is -1, 0 or 1.Comment: 15 page

Column Generation Algorithms for Nonlinear Optimization II: Numerical Investigations

García et al. present a class of column generation (CG) algorithms for nonlinear programs. Its main motivation from a theoretical viewpoint is that under some circumstances, finite convergence can be achieved, in much the same way as for the classic simplicial decomposition method; the main practical motivation is that within the class there are certain nonlinear column generation problems that can accelerate the convergence of a solution approach which generates a sequence of feasible points. This algorithm can, for example, accelerate simplicial decomposition schemes by making the subproblems nonlinear. This paper complements the theoretical study on the asymptotic and finite convergence of these methods given in [1] with an experimental study focused on their computational efficiency. Three types of numerical experiments are conducted. The first group of test problems has been designed to study the parameters involved in these methods. The second group has been designed to investigate the role and the computation of the prolongation of the generated columns to the relative boundary. The last one has been designed to carry out a more complete investigation of the difference in computational efficiency between linear and nonlinear column generation approaches. In order to carry out this investigation, we consider two types of test problems: the first one is the nonlinear, capacitated single-commodity network flow problem of which several large-scale instances with varied degrees of nonlinearity and total capacity are constructed and investigated, and the second one is a combined traffic assignment mode

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274