Global relaxation-based LP-Newton method for multiple hyperparameter selection in support vector classification with feature selection

Support vector classification (SVC) is an effective tool for classification tasks in machine learning. Its performance relies on the selection of appropriate hyperparameters. This paper focuses on optimizing the regularization hyperparameter C and determining feature bounds for feature selection within SVC, leading to a potentially large hyperparameter space. It is well known in machine learning that this can lead to the so-called curse of dimensionality. To address this challenge of multiple hyperparameter selection, the problem is formulated as a bilevel optimization problem, which is then transformed into a mathematical program with equilibrium constraints (MPEC). Our primary contributions are twofold. First, we establish the satisfaction of the MPEC-MFCQ for our problem reformulation. Furthermore, we introduce a novel global relaxation-based linear programming (LP)-Newton method (GRLPN) for solving this problem and provide corresponding convergence results. Typically, in global relaxation methods for MPECs, the algorithm for the corresponding subproblem is treated as a black box. Possibly for the first time in the literature, the subproblem is specifically studied in detail. Numerical experiments demonstrate GRLPN's superiority in efficiency and accuracy over both grid search and traditional global relaxation methods solved using the well-known nonlinear programming solver SNOPT.