We apply nonsmooth optimization techniques to classification problems, with particular reference to the transductive support vector machine (TSVM) approach, where the considered decision function is nonconvex and nondifferentiable, hence difficult to minimize. We present some numerical results obtained by running the proposed method on some standard test problems drawn from the binary classification literature.

The literature in the area of the semi-supervised binary classification has demonstrated that useful information can be gathered not only from those samples whose class membership is known in advance, but also from the unlabelled ones. In fact, in the support vector machine, semi-supervised models with both labelled and unlabelled samples contribute to the definition of an appropriate optimization model for finding a good quality separating hyperplane. In particular, the optimization approaches which have been devised in this context are basically of two types: a mixed integer linear programming problem, and a continuous optimization problem characterized by an objective function which is nonsmooth and nonconvex. Both such problems are hard to solve whenever the number of the unlabelled points increases. In this article, we present a data preprocessing technique which has the objective of reducing the number of unlabelled points to enter the computational model, without worsening too much the classification performance of the overall process. The approach is based on the concept of separating sets and can be implemented with a reasonable computational effort. The results of the numerical experiments on several benchmark datasets are also reported.

We present a bundle method for convex nondifferentiable minimization where the model is a piecewise-quadratic convex approximation of the objective function. Unlike standard bundle approaches, the model only needs to support the objective function from below at a properly chosen (small) subset of points, as opposed to everywhere. We provide the convergence analysis for the algorithm, with a general form of master problem which combines features of trust region stabilization and proximal stabilization, taking care of all the important practical aspects such as proper handling of the proximity parameters and the bundle of information. Numerical results are also reported.