  Support vector classification (SVC) is an effective tool for classification
tasks in machine learning. Its performance relies on the selection of
appropriate hyperparameters. In this paper, our focus is on identifying the
optimal value for the regularization hyperparameter C, as well as determining
the bounds on the features in a SVC problem. This implies that the number of
hyperparameters in our SVC can potentially be very large. It is very well-known
in machine learning that this could 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 first contribution involves proving the fulfillment of a
Mangasarian-Fromovitz constraint qualification tailored to the latter
reformulation of the problem. Furthermore, we introduce a novel linear
programming (LP)-Newton-based global relaxation 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 blackbox. Possibly for the first time in the literature, the
subproblem is specifically studied in detail. Numerical experiments
substantiate the superiority of GRLPN over grid search and the global
relaxation solved by the well-known nonlinear programming solver SNOPT.

Optimization and Control