In this paper, we study the embedded feature selection problem in linear Support Vector Machines (SVMs), in which a cardinality constraint is employed, leading to an interpretable classification model. The problem is NP-hard due to the presence of the cardinality constraint, even though the original linear SVM amounts to a problem solvable in polynomial time. To handle the hard problem, we first introduce two mixed-integer formulations for which novel semidefinite relaxations are proposed. Exploiting the sparsity pattern of the relaxations, we decompose the problems and obtain equivalent relaxations in a much smaller cone, making the conic approaches scalable. To make the best usage of the decomposed relaxations, we propose heuristics using the information of its optimal solution. Moreover, an exact procedure is proposed by solving a sequence of mixed-integer decomposed semidefinite optimization problems. Numerical results on classical benchmarking datasets are reported, showing the efficiency and effectiveness of our approach.

翻译：本文研究线性支持向量机中的嵌入式特征选择问题，该问题通过引入基数约束以获得可解释的分类模型。尽管原始线性支持向量机可在多项式时间内求解，但基数约束的存在使得该问题成为NP难问题。为处理这一难题，我们首先提出两种混合整数规划形式，并为其构建了新颖的半定松弛模型。通过利用松弛模型的稀疏特性，我们对问题进行分解，并在维度显著缩小的锥空间中获得了等价松弛，从而使锥优化方法具备可扩展性。为充分利用分解后的松弛模型，我们基于其最优解信息设计了启发式算法。此外，通过求解一系列混合整数分解半定优化问题，提出了精确求解流程。在经典基准数据集上的数值实验结果验证了所提方法的高效性与有效性。