On convex lower-level black-box constraints in bilevel optimization with an application to gas market models with chance constraints

Bilevel optimization is an increasingly important tool to model hierarchical decision making. However, the ability of modeling such settings makes bilevel problems hard to solve in theory and practice. In this paper, we add on the general difficulty of this class of problems by further incorporating convex black-box constraints in the lower level. For this setup, we develop a cutting-plane algorithm that computes approximate bilevel-feasible points. We apply this method to a bilevel model of the European gas market in which we use a joint chance constraint to model uncertain loads. Since the chance constraint is not available in closed form, this fits into the black-box setting studied before. For the applied model, we use further problem-specific insights to derive bounds on the objective value of the bilevel problem. By doing so, we are able to show that we solve the application problem to approximate global optimality. In our numerical case study we are thus able to evaluate the welfare sensitivity in dependence of the achieved safety level of uncertain load coverage.

Automated summary

Bilevel optimization is a powerful tool for modeling hierarchical decision making, but it is challenging to solve in theory and practice. This paper addresses this challenge by incorporating convex black-box constraints in the lower level and developing a cutting-plane algorithm to find approximate bilevel-feasible points. The method is applied to a bilevel model of the European gas market with joint chance constraints, and problem-specific insights are used to derive bounds on the objective value. The numerical case study evaluates the welfare sensitivity based on the achieved safety level of uncertain load coverage.