Introduction to linear and nonlinear programming

Over the past decade, the field of finite-dimensional variational inequality and complementarity problems has seen a rapid development in its theory of existence, uniqueness and sensitivity of solution(s), in the theory of algorithms, and in the application of these techniques to transportation planning, regional science, socio-economic analysis, energy modeling, and game theory. This paper provides a state-of-the-art review of these developments as well as a summary of some open research topics in this growing field.

Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications

In network design, the gap between theory and practice is woefully broad. This book narrows it, comprehensively and critically examining current network design models and methods. You will learn where mathematical modeling and algorithmic optimization have been under-utilized. At the opposite extreme, you will learn where they tend to fail to contribute to the twin goals of network efficiency and cost-savings. Most of all, you will learn precisely how to tailor theoretical models to make them as useful as possible in practice. Throughout, the authors focus on the traffic demands encountered in the real world of network design. Their generic approach, however, allows problem formulations and solutions to be applied across the board to virtually any type of backbone communication or computer network. For beginners, this book is an excellent introduction. For seasoned professionals, it provides immediate solutions and a strong foundation for further advances in the use of mathematical modeling for network design. (Less)

Routing, Flow, And Capacity Design In Communication And Computer Networks

Network Optimization: Continuous and Discrete Models

We describe a model for the transit assignment problem with a fixed set of transit lines The traveler chooses the strategy that allows him or her to reach his or her destination at minimum expected cost First we consider the case in which no congestion effects occur For the special case in which the waiting time at a stop depends only on the combined frequency, the problem is formulated as a linear programming problem of a size that increases linearly with the network size A label-setting algorithm is developed that solves the latter problem in polynomial time Nonlinear cost extensions of the model are considered as well

Optimal strategies: A new assignment model for transit networks

We describe a model for estimating an origin-destination matrix from an observed sample matrix, when the volumes on a subset of the links of the network and/or the total productions and attractions of the zones are known. The elements of the observed sample matrix are assumed to be integers that are obtained from independent Poisson distributions with unknown means. A maximum likelihood model is formulated to estimate these means, yielding an estimation of the “true” origin- destination matrix which is consistent with the observed link volumes. Conditions for existence and uniqueness of a solution are discussed. A solution algorithm based on the cyclic coordinate descent method is developed and its convergence properties are analyzed. The special case of the matrix estimation problem, in which marginal totals are given instead of link volumes, is considered separately; a numerical example is used to illustrate the problem. Using results about the asymptotic behavior of the distribution of the likelihood function, tests may be derived that allow statistical inferences on the consistency of the available data. Finally, an extension of the model is studied in which the observed volumes are Poisson-distributed as well.

A maximum likelihood model for estimating origin-destination matrices

The widely used BPR volume-delay functions have some inherent drawbacks. A set of conditions is developed which a “well behaved” volume delay function should satisfy. This leads to the definition of a new class of functions named conical volume-delay functions, due to their geometrical interpretation as hyperbolic conical sections. It is shown that these functions satisfy all conditions set forth and, thus, constitute a viable alternative to the BPR type functions.

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Technical Note—Conical Volume-Delay Functions

The widely used BPR volume-delay functions have some inherent drawbacks. A set of conditions is developed which a “well behaved” volume delay function should satisfy. This leads to the definition of a new class of functions named conical volume-delay functions , due to their geometrical interpretation as hyperbolic conical sections. It is shown that these functions satisfy all conditions set forth and, thus, constitute a viable alternative to the BPR type functions.

/pdf/conical-volume-delay-functions-4qo4fetsvv.pdf

Conical volume-delay functions

We provide a sufficient condition for the convergence of diagonalization algorithms for equilibrium traffic assignment problems with asymmetric Jacobian matrix B(v) of the link user cost mapping s(v) of the flow v. When |I−D(v ∗ ) − 1 2 B(v ∗ ) D(v ∗ − 1 2 | 2 , where D(v ∗ ) > 0 is the diagonal of B(v∗) and v∗ is the equilibrium flow, we demonstrate a local convergence theorem for nonlinear cost functions. The implication of this result for practical applications of the model are outlined.

Heinz Spiess

Papers

Optimal strategies: A new assignment model for transit networks

A maximum likelihood model for estimating origin-destination matrices

Technical Note—Conical Volume-Delay Functions

Conical volume-delay functions

The convergence of diagonalization algorithms for asymmetric network equilibrium problems