Patrice Marcotte

An overview of bilevel optimization

Abstract This paper is devoted to bilevel optimization, a branch of mathematical programming of both practical and theoretical interest. Starting with a simple example, we proceed towards a general formulation. We then present fields of application, focus on solution approaches, and make the connection with MPECs (Mathematical Programs with Equilibrium Constraints).

Bilevel programming: A survey

Abstract.This paper provides an introductory survey of a class of optimization problems known as bilevel programming. We motivate this class through a simple application, and then proceed with the general formulation of bilevel programs. We consider various cases (linear, linear-quadratic, nonlinear), describe their main properties and give an overview of solution approaches.

On the relationship between Nash-Cournot and Wardrop equilibria

A noncooperative game is formulated on a transportation network with congestion. The players are associated with origin-destination pairs, and are facing demand functions at their respective destination nodes. A Nash-Cournot equilibrium is defined and conditions for existence and uniqueness of this solution are provided. The asymptotic behavior of the Nash-Cournot equilibrium is then shown to yield (under appropriate assumptions) a total flow vector corresponding to a Wardrop equilibrium.

Network design problem with congestion effects: A case of bilevel programming

Recently much attention has been focused on multilevel programming, a branch of mathematical programming that can be viewed either as a generalization of min-max problems or as a particular class of Stackelberg games with continuous variables. The network design problem with continuous decision variables representing link capacities can be cast into such a framework. We first give a formal description of the problem and then develop various suboptimal procedures to solve it. Worst-case behaviour results concerning the heuristics, as well as numerical results on a small network, are presented.

Transit Equilibrium Assignment: A Model and Solution Algorithms

This paper proposes a model for the transit equilibrium assignment problem (TEAP) and develops two algorithms for its solution. The behavior of the transit users is modeled by using the concept for hyperpaths on an appropriate network that is obtained from the road network and the transit lines by a transformation that makes explicit the walk, wait, in-vehicle, transfer, and alight arcs. The TEAP is stated and formulated as a variational inequality problem, in the space of hyperpath flows, and then solved by the linearized Jacobi method and the projection method. The global convergence of these two algorithms for strongly monotone arc cost mappings is proven and the implementation of the algorithms and computational experiments are presented as well.

Advances in the Continuous Dynamic Network Loading Problem

The continuous dynamic network loading problem (CDNLP) consists in determining, on a congested network, time-dependent arc volumes, together with arc and path travel times, given the time varying path flow departure rates over a finite time horizon. This problem constitutes an intrinsic part of the dynamic traffic assignment problem. In this paper, the authors present a formulation of the CDNLP where travel delays may be nonlinear functions of arc traffic volumes. They show, under a boundedness condition, that there exists a unique solution to the problem and propose for its solution a finite-step algorithm. Some computational results are reported for a discretized version of the algorithm.(A)

EQUILIBRIUM and advanced transportation modelling

Preface. Introduction P. Marcotte, Sang Nguyen. 1. Microscopic Traffic Simulation J. Barcelo, et al. 2. Activity Based Travel Demand Model Systems M.E. Ben-Akiva, J.L. Bowman. 3. Passenger Assignment in Congested Transit Networks: A Historical Perspective B. Bouzaiene-Ayari, et al. 4. Long-term Advances in the State of the Art of Travel Forecasting Methods D. Boyce. 5. Stochastic Assignment to Transportation Networks: Models and Algorithms G.E. Cantarella, E. Cascetta. 6. Solving Congestion Toll Pricing Models D.W. Hearn, M.V. Ramana. 7. Side Constrained Traffic Equilibrium Models - Traffic Management Through Link Tolls T. Larsson, M. Patriksson. 8. Multicriteria Assignment Modeling: Making Explicit the Determinants of Mode or Path Choice F.M. Leurent. 9. Hyperpath Formulations of Traffic Assignment Problems P. Marcotte, Sang Nguyen. 10. Network Equilibria and Disequilibria A. Nagurney, Ding Zhang. 11. Shortest Path Algorithms in Transportation Models: Classical and Innovative Aspects S. Pallottino, M.G. Scutella. 12. Bilevel and Other Modelling Approaches to Urban Traffic Management and Control M.J. Smith, et al.

A Trust-Region Method for Nonlinear Bilevel Programming: Algorithm and Computational Experience

We consider the approximation of nonlinear bilevel mathematical programs by solvable programs of the same type, i.e., bilevel programs involving linear approximations of the upper-level objective and all constraint-defining functions, as well as a quadratic approximation of the lower-level objective. We describe the main features of the algorithm and the resulting software. Numerical experiments tend to confirm the promising behavior of the method.

A note on a globally convergent Newton method for solving monotone variational inequalities

It is well-known (see Pang and Chan [8]) that Newtons method, applied to strongly monotone variational inequalities, is locally and quadratically convergent. In this paper we show that Newtons method yields a descent direction for a non-convex, non-differentiable merit function, even in the absence of strong monotonicity. This result is then used to modify Newtons method into a globally convergent algorithm by introducing a linesearch strategy. Furthermore, under strong monotonicity (i) the optimal face is attained after a finite number of iterations, (ii) the stepsize is eventually fixed to the value one, resulting in the usual Newton step. Computational results are presented.

Some comments of Wolfe's `away step'

We give a detailed proof, under slightly weaker conditions on the objective function, that a modified Frank-Wolfe algorithm based on Wolfes ‘away step’ strategy can achieve geometric convergence, provided a strict complementarity assumption holds.