Optimization meets Machine Learning: An Exact Algorithm for
  Semi-Supervised Support Vector Machines
Veronica Piccialli Jan Schwiddessen Antonio M. Sudoso
Support vector machines (SVMs) are well-studied supervised learning models
for binary classification. In many applications, large amounts of samples can
be cheaply and easily obtained. What is often a costly and error-prone process
is to manually label these instances. Semi-supervised support vector machines
(S3VMs) extend the well-known SVM classifiers to the semi-supervised approach,
aiming at maximizing the margin between samples in the presence of unlabeled
data. By leveraging both labeled and unlabeled data, S3VMs attempt to achieve
better accuracy and robustness compared to traditional SVMs. Unfortunately, the
resulting optimization problem is non-convex and hence difficult to solve
exactly. In this paper, we present a new branch-and-cut approach for S3VMs
using semidefinite programming (SDP) relaxations. We apply optimality-based
bound tightening to bound the feasible set. Box constraints allow us to include
valid inequalities, strengthening the lower bound. The resulting SDP relaxation
provides bounds significantly stronger than the ones available in the
literature. For the upper bound, instead, we define a local search exploiting
the solution of the SDP relaxation. Computational results highlight the
efficiency of the algorithm, showing its capability to solve instances with a
number of data points 10 times larger than the ones solved in the literature.