A parametric programming approach to bilevel optimisation with lower-level variables in the upper level

Henrik C. Bylling, Steven A. Gabriel and Trine K. Boomsma

Journal of the Operational Research Society, 2020, vol. 71, issue 5, 846-865

Abstract: This paper examines linearly constrained bilevel programming problems in which the upper-level objective function depends on both the lower-level primal and dual optimal solutions. We parametrize the lower-level solutions and thereby the upper-level objective function by the upper-level variables and argue that it may be non-convex and even discontinuous. However, when the upper-level objective is affine in the lower-level primal optimal solution, the parametric function is piece-wise linear. We show how this property facilitates the application of parametric programming and demonstrate how the approach allows for decomposition of a separable lower-level problem. When the upper-level objective is bilinear in the lower-level primal and dual optimal solutions, we also provide an exact linearisation method that reduces the bilevel problem to a single-level mixed-integer linear programme (MILP). We assess the performance of the parametric programming approach on two case studies of strategic investment in electricity markets and benchmark against state-of-the-art MILP and non-linear solution methods for bilevel optimisation problems. Preliminary results indicate substantial computational advantages over several standard solvers, especially when the lower-level problem separates into a large number of subproblems. Furthermore, we show that the parametric programming approach succeeds in solving problems to global optimality for which standard methods can fail.

Date: 2020
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DOI: 10.1080/01605682.2019.1590132

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