Improving Directions in Mixed Integer Bilevel Linear Optimization

We consider the central role of improving directions in solution methods for mixed integer bilevel linear optimization problems (MIBLPs). Current state-of-the-art methods for solving MIBLPs employ the branch-and-cut framework originally developed for solving mixed integer linear optimization problems. This approach relies on oracles for two kinds of subproblems: those for checking whether a candidate pair of leader's and follower's decisions is bilevel feasible, and those required for generating valid inequalities. Typically, these two types of oracles are managed separately, but in this work, we explore their close connection and propose a solution framework based on solving a single type of subproblem: determining whether there exists a so-called improving feasible direction for the follower's problem. Solution of this subproblem yields information that can be used both to check feasibility and to generate strong valid inequalities. Building on prior works, we expose the foundational role of improving directions in enforcing the follower's optimality condition and extend a previously known hierarchy of optimality-based relaxations to the mixed-integer setting, showing that the associated relaxed feasible regions coincide exactly with the closure associated with intersection cuts derived from improving directions. Numerical results with an implementation using a modified version of the open source solver MibS show that this approach can yield practical improvements.