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.

翻译：本文探讨了改进方向在混合整数双层线性优化问题求解方法中的核心作用。当前最先进的MIBLP求解方法采用最初为混合整数线性优化问题开发的"分支-切割"框架。该方法依赖于两类子问题的求解机制：用于验证领导者与跟随者决策候选对是否满足双层可行性的验证机制，以及用于生成有效不等式的生成机制。通常这两类机制独立运作，但本研究通过揭示其内在关联，提出了一种基于单一类型子问题求解的框架：判断跟随者问题是否存在所谓的可行改进方向。该子问题的求解结果既能用于可行性验证，又能生成强有效不等式。基于前人研究，我们揭示了改进方向在保证跟随者最优性条件中的基础性作用，并将已知的最优性松弛层次结构扩展至混合整数情形，证明相关松弛可行域与基于改进方向的交割闭包完全一致。采用开源求解器MibS改进版本的数值实验表明，该方法能带来实际性能提升。