A trust-region method for nonlinear bilevel optimisation problems with an application in transportation

Abstract

A bilevel optimisation problem is an optimisation problem which has a second optimisation problem embedded in its constraints. It aims to model problems and decision processes that are hierarchical, which are problem structures that occur frequently in real-life. Thus, due to the wide range of applications of bilevel problems, there is a strong motivation to solve them. The aim of this thesis is to develop an approach to solving bilevel programs by utilising the less commonly used optimal value reformulation. The work can be split into two main contributions. First, a novel trust-region approach to solving nonlinear bilevel problems is proposed, which solves an exact penalisation of the optimal value reformulation. Second, an application of bilevel programming to the London congestion pricing problem is explored, investigating the application of the proposed trust-region method to solve a bilevel formulation of the road pricing problem. One of the most common approaches to solving a bilevel program is to first transform the problem into a single level program. The most popular way of doing so is by replacing the lower level problem by its Karush-Kuhn Tucker conditions. That being said, the reformulation requires strong assumptions on the bilevel program for it to be equivalent to the original problem. An alternative method to transform the problem into a single level problem is to use the optimal value function of the lower level problem. This problem is known to be fully equivalent. However, due to the difficulties in solving it, approaches in the literature that utilise this reformulation are relatively scarce. Under a regularity condition known as the partial calmness condition, an exact penalty problem can be built from the optimal value reformulation. The first contribution of this thesis is the investigation of solving this exact penalty problem to find local solutions of the associated bilevel problem. A novel trust-region algorithm is proposed to solve it, and convergence analysis is explored. The implementation and performance of the algorithm is investigated via extensive numerical experiments on a large test set of nonlinear bilevel problems. This provides strong numerical results that validate the approach for solving bilevel problems. Based on the results and analysis, the performance and limitations of the algorithms are discussed. The second contribution of this thesis is exploring the application of the aforementioned trust-region method on the bilevel optimisation formulation of the road pricing problem. Road pricing is a method used by governments to alleviate congestion in an overcrowded network. The problem has a hierarchical structure, and therefore naturally forms as a bilevel program. We investigate a case study of the problem to the London congestion zone charge: a fixed cordon road pricing scheme implemented in the center of London. Although successful on initial implementation in 2003, congestion in the city has returned to pre-charge levels. Due to recent advances in technology, the Mayor of London is looking to update the congestion charge to a more sophisticated tolling scheme that can charge for distance, time, emissions or other factors. A formulation of the London congestion problem as a bilevel program is presented, which considers the aims and objectives set out by the current Mayor of London. We then show how the trust-region algorithm can be applied to solve a simplified form of the road pricing model commonly seen in the literature. This is a novel approach to the problem, solving a single level continuous exact penalty problem to find local solutions of the road pricing bilevel model. The performance of the algorithm is tested and verified numerically on three network examples of varying sizes, and the efficiency and robustness of the algorithm is assessed.

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ORCID for Joerg Fliege: orcid.org/0000-0002-4459-5419

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