Steinborn-Busse, Karl-Matthias
(2022)
Formulations and solution methods for a mixed-integer non-linear bilevel pricing problem.
*University of Southampton, Doctoral Thesis*, 138pp.

## Abstract

Since the turn of the 21st century, interest in bilevel optimisation has been rapidly increasing. With a vast array of bilevel applications in energy and transport industries, interest has always existed, however, the solution methods, combined with our computational powers, meant many sizeable bilevel problems were out of reach. Most solution methods focused on very simplistic cases, continuous variables with linear constraints, whereas many problems of interest included non-linear constraints along with integer variables. In this thesis, we study a Mixed-Integer Non-Linear Bilevel Problem, MINLBP, based upon a commodity pricing problem. The objective of the followers is to purchase a set of commodities, that satisfy some combinatorial constraints, at the minimum cost while the leader wishes to maximise their profit by introducing a taxation to commodities. We shall present three formulations; unit supply and single follower, unit supply and multiple followers and non-unit supply and multiple followers. Unit and non-unit supply relates to the quantity of each commodity available for both the leader and follower to purchase. As it shall be shown, the unit and non-unit supply scenarios must be treated differently as it drastically affects the reaction of the follower. With the single supply case, should a feasible follower reaction include commodity j, then they shall purchase that commodity. However, in the non-unit supply case, the follower can select the cheapest version of commodity j available to them. Thus, should the leader have applied a taxation to one version of j, then the follower would purchase an untaxed version instead. As a result we observe that, as the result of a max operator in both the leader and follower objective functions, we can reformulate the problem, using two methods that we call the y¯ and γ reformulations. Following this, we shall linearise our formulations to produce a Mixed-Integer Linear Problem, MILP, using McCormick envelopes, along with binary expansions when necessary. We then address various solution methods to be used in conjunction with the formulations described. We discuss how a simple cutting-plane approach, used within a branch-and-cut framework can easily be implemented for the unit supply case, but cannot be directly applied for the more general non-unit supply case. This is a result of the followers feasible region no longer being completely independent of the leaders variables. Therefore, we demonstrate four solution methods for the general case; two of these use a variable generation approach to introduce variables, which ensure the value function constraint is only active for the necessary sections of the feasible region, with the latter two using a sophisticated branching strategy to achieve the same results without the need for any additional variables. We also present further solution methods focused on solving instances where the followers are restricted to a subset of responses. The first solution method is split into two parts. The first of which iteratively solves the case where the followers must respond from a set Y, adding reactions to Y with every bilevel infeasible solution. The second v stage then solves the case where we assume that the followers respond with a solution outside of Y. Following this, we shall present a branching strategy which aims to encompass this solution method within a single Branch-and-Cut framework. Three sets of instances are generated to compare the performances of both the formulations and solution methods. Each focus their attention on certain parameters of the bilevel problem including the number of commodities, the maximum taxation and the leaders budget

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