Tightening big Ms in integer programming formulations for support vector machines with ramp loss

• Published in: European journal of operational research Vol. 286; no. 1; pp. 84 - 100
• Main Authors: Baldomero-Naranjo, Marta, Martínez-Merino, Luisa I., Rodríguez-Chía, Antonio M.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.10.2020
• Subjects: Indicator constraints; Location; Mixed integer programming; Ramp loss model; Support vector machine; Mixed integer programming; Support vector machine; Indicator constraints; Ramp loss model; Location
• ISSN: 0377-2217; 1872-6860
• DOI: 10.1016/j.ejor.2020.03.023

•Support vector machines with ramp loss models are analyzed.•Strategies for tightening the values of big M parameters of the model are proposed.•The strategies are focused on ℓ1-norm and ℓ2-norm cases.•Linear, quadratic and Lagragian relaxation models are solved in the strategies.•The performance of these strategies are tested in simulated and real-life datasets. This paper considers various models of support vector machines with ramp loss, these being an efficient and robust tool in supervised classification for the detection of outliers. The exact solution approaches for the resulting optimization problem are of high demand for large datasets. Hence, the goal of this paper is to develop algorithms that provide efficient methodologies to exactly solve these optimization problems. These approaches are based on three strategies for obtaining tightened values of the big M parameters included in the formulation of the problem. Two of them require solving a sequence of continuous problems, while the third uses the Lagrangian relaxation to tighten the bounds. The proposed resolution methods are valid for the ℓ1-norm and ℓ2-norm ramp loss formulations. They were tested and compared with existing solution methods in simulated and real-life datasets, showing the efficiency of the developed methodology.