In this paper, we propose a new mathematical optimization model for multiclass classification based on arrangements of hyperplanes. Our approach preserves the core support vector machine (SVM) paradigm of maximizing class separation while minimizing misclassification errors, and it is computationally more efficient than a previous formulation. We present a kernel-based extension that allows it to construct nonlinear decision boundaries. Furthermore, we show how the framework can naturally incorporate alternative geometric structures, including classification trees, $\ell_p$-SVMs, and models with discrete feature selection. To address large-scale instances, we develop a dynamic clustering matheuristic that leverages the proposed MIP formulation. Extensive computational experiments demonstrate the efficiency of the proposed model and dynamic clustering heuristic, and we report competitive classification performance on both synthetic datasets and real-world benchmarks from the UCI Machine Learning Repository, comparing our method with state-of-the-art implementations available in scikit-learn.

翻译：本文提出了一种基于超平面排列的多类分类新数学优化模型。该方法保留了支持向量机（SVM）最大化类别间隔并最小化误分类误差的核心范式，且计算效率优于先前提出的模型。我们提出了基于核函数的扩展，使其能够构建非线性决策边界。此外，我们展示了该框架如何自然地融入替代几何结构，包括分类树、$\ell_p$-支持向量机以及具有离散特征选择的模型。为处理大规模问题实例，我们开发了一种利用所提出的混合整数规划（MIP）公式的动态聚类元启发式算法。大量计算实验证明了所提模型及动态聚类启发式算法的效率；通过在合成数据集和来自UCI机器学习知识库的真实基准数据集上进行测试，并将我们的方法与scikit-learn中现有的先进实现进行比较，我们报告了具有竞争力的分类性能。