Inverse Mixed Integer Optimization: An Interior Point Perspective
Samir Elhedhli, Göksu Ece Okur
TL;DR
This work addresses inverse optimization for forward problems with mixed-integer structure by introducing an interior-point–inspired framework that treats mixed-integer solutions as interior analytic centers. The authors develop LP-based reformulations that tie the mixed-integer point to an LP relaxation, quantify the LP-gap, and formulate two practical models: a tolerance model and a bi-objective model, both solvable as linear programs. They demonstrate through extensive MIPLIB 2017 testing that these models produce high-quality cost vectors and near-optimal mixed-integer solutions with very fast CPU times, often outperforming the state-of-the-art trust-region cutting plane method. To guarantee optimality of the given solution, they supplement the proposed models with a classical cutting-plane approach, yielding a robust, scalable framework for inverse mixed-integer optimization with strong practical applicability.
Abstract
We propose a novel solution framework for inverse mixed-integer optimization based on analytic center concepts from interior point methods. We characterize the optimality gap of a given solution, provide structural results, and propose models that can efficiently solve large problems. First, we exploit the property that mixed-integer solutions are primarily interior points that can be modeled as weighted analytic centers with unique weights. We then demonstrate that the optimality of a given solution can be measured relative to an identifiable optimal solution to the linear programming relaxation. We quantify the absolute optimality gap and pose the inverse mixed-integer optimization problem as a bi-level program where the upper-level objective minimizes the norm to a given reference cost, while the lower-level objective minimizes the absolute optimality gap to an optimal linear programming solution. We provide two models that address the discrepancies between the upper and lower-level problems, establish links with noisy and data-driven optimization, and conduct extensive numerical testing. We find that the proposed framework successfully identifies high-quality solutions in rapid computational times. Compared to the state-of-the-art trust region cutting plane method, it achieves optimal cost vectors for 95% and 68% of the instances within optimality gaps of e-2 and e-5, respectively, without sacrificing the relative proximity to the nominal cost vector. To ensure the optimality of the given solution, the proposed approach is complemented by a classical cutting plane method. It is shown to solve instances that the trust region cutting plane method could not successfully solve as well as being in very close proximity to the nominal cost vector.