Introduction to linear and nonlinear programming

Routing problems appear frequently when dealing with the operation of communication or transportation networks. Among them, the message routing problem plays a determinant role in the optimization of network performance. Much of the motivation for this work comes from this problem which is shown to belong to the class of nonlinear convex multicommodity flow problems. This paper emphasizes the message routing problem in data networks, but it includes a broader literature overview of convex multicommodity flow problems. We present and discuss the main solution techniques proposed for solving this class of large-scale convex optimization problems. We conduct some numerical experiments on the message routing problem with four different techniques.

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A Survey of Algorithms for Convex Multicommodity Flow Problems

This paper addresses two second-best toll pricing problems, one with fixed and the other with elastic travel demands, as mathematical programs with equilibrium constraints. Several equivalent nonlinear programming formulations for the two problems are discussed. One formulation leads to properties that are of interest to transportation economists. Another produces an algorithm that is capable of solving large problems and easy to implement with existing software for linear and nonlinear programming problems. Numerical results using transportation networks from the literature are also presented.

An MPEC approach to second-best toll pricing

https://link.springer.com/content/pdf/10.1007%2Fs13676-013-0021-5.pdf

A network equilibrium analysis on destination, route and parking choices with mixed gasoline and electric vehicular flows

We present versions of the Frank-Wolfe method for linearly constrained convex programs, in which consecutive search directions are made conjugate. Preliminary computational studies in a MATLAB environment applying pure Frank-Wolfe, conjugate direction Frank-Wolfe CFW, bi-conjugate Frank-Wolfe BFW, and “partanized” Frank-Wolfe methods to some classical Traffic Assignment Problems show that CFW and BFW compare favorably to the other methods. This spurred a more detailed study, comparing our methods to an origin-based algorithm. This study indicates that our methods are competitive for accuracy requirements ensuring link flow stability. We also show that CFW is globally convergent. We further point at independent studies by other researchers that show that our methods compare favorably with recent bush-based and gradient projection algorithms on computers with several cores.

/pdf/the-stiff-is-moving-conjugate-direction-frank-wolfe-methods-35mjf0cyc3.pdf

The Stiff Is Moving---Conjugate Direction Frank-Wolfe Methods with Applications to Traffic Assignment*

The PARTAN variant of the linear approximation method is adapted for solving the network equilibrium problem. A simple and efficient algorithm is stated. Its properties are analyzed by using algebraic and geometric approaches. Its computational efficiency on small and large scale problems is compared to that of the linear approximation method.

J Guelat

Papers

An efficient implementation of the "partan" variant of the linear approximation method for the network equilibrium problem