Multi-parametric global optimization approach for tri-level mixed-integer linear optimization problems

When generating multirow intersection cuts for mixed-integer linear optimization problems, an important practical question is deciding which intersection cuts to use. Even when restricted to cuts that are facet defining for the corner relaxation, the number of potential candidates is still very large, especially for instances of large size. In this paper, we introduce a subset of intersection cuts based on the infinity norm that is very small, works for relaxations having arbitrary number of rows and, unlike many subclasses studied in the literature, takes into account the entire data from the simplex tableau. We describe an algorithm for generating these inequalities and run extensive computational experiments in order to evaluate their practical effectiveness in real-world instances. We conclude that this subset of inequalities yields, in terms of gap closure, around 50% of the benefits of using all valid inequalities for the corner relaxation simultaneously, but at a small fraction of the computational cost, and with a very small number of cuts. Summary of Contribution: Cutting planes are one of the most important techniques used by modern mixed-integer linear programming solvers when solving a variety of challenging operations research problems. The paper advances the state of the art on general-purpose multirow intersection cuts by proposing a practical and computationally friendly method to generate them.

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Collision Avoidance in Air Traffic Management: A Mixed-Integer Linear Optimization Approach

IEEE Transactions on Intelligent Transportation Systems ◽

10.1109/tits.2010.2061971 ◽

2011 ◽

Vol 12 (1) ◽

pp. 47-57 ◽

Cited By ~ 60

Author(s):

Antonio Alonso-Ayuso ◽

Laureano F. Escudero ◽

F. Javier Martin-Campo

Keyword(s):

Collision Avoidance ◽

Traffic Management ◽

Linear Optimization ◽

Air Traffic Management ◽

Air Traffic ◽

Mixed Integer ◽

Optimization Approach ◽

Mixed Integer Linear Optimization ◽

Integer Linear Optimization

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Inverse Mixed Integer Optimization: Polyhedral Insights and Trust Region Methods

INFORMS Journal on Computing ◽

10.1287/ijoc.2021.1138 ◽

2022 ◽

Author(s):

Merve Bodur ◽

Timothy C. Y. Chan ◽

Ian Yihang Zhu

Keyword(s):

Optimization Problems ◽

Linear Optimization ◽

Trust Region ◽

Inverse Optimization ◽

Mixed Integer ◽

Integer Optimization ◽

Solution Algorithms ◽

Linear Optimization Problems ◽

Mixed Integer Linear Optimization ◽

Plane Solution

Inverse optimization—determining parameters of an optimization problem that render a given solution optimal—has received increasing attention in recent years. Although significant inverse optimization literature exists for convex optimization problems, there have been few advances for discrete problems, despite the ubiquity of applications that fundamentally rely on discrete decision making. In this paper, we present a new set of theoretical insights and algorithms for the general class of inverse mixed integer linear optimization problems. Specifically, a general characterization of optimality conditions is established and leveraged to design new cutting plane solution algorithms. Through an extensive set of computational experiments, we show that our methods provide substantial improvements over existing methods in solving the largest and most difficult instances to date.

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DACE: Distribution-Aware Counterfactual Explanation by Mixed-Integer Linear Optimization

Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence ◽

10.24963/ijcai.2020/395 ◽

2020 ◽

Author(s):

Kentaro Kanamori ◽

Takuya Takagi ◽

Ken Kobayashi ◽

Hiroki Arimura

Keyword(s):

Cost Function ◽

Empirical Data ◽

Linear Optimization ◽

Data Distribution ◽

Mixed Integer ◽

Optimization Approach ◽

Mixed Integer Linear Optimization ◽

Post Hoc ◽

New Framework ◽

Integer Linear Optimization

Counterfactual Explanation (CE) is one of the post-hoc explanation methods that provides a perturbation vector so as to alter the prediction result obtained from a classifier. Users can directly interpret the perturbation as an "action" for obtaining their desired decision results. However, an action extracted by existing methods often becomes unrealistic for users because they do not adequately care about the characteristics corresponding to the empirical data distribution such as feature-correlations and outlier risk. To suggest an executable action for users, we propose a new framework of CE for extracting an action by evaluating its reality on the empirical data distribution. The key idea of our proposed method is to define a new cost function based on the Mahalanobis' distance and the local outlier factor. Then, we propose a mixed-integer linear optimization approach to extracting an optimal action by minimizing our cost function. By experiments on real datasets, we confirm the effectiveness of our method in comparison with existing methods for CE.

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