Large-scale optimization: new active-set methods and application in unsupervised learning

In this thesis, new methods for large-scale non-linear optimization are presented. In particular, an active-set algorithm for bound-constrained optimization is first proposed, characterized by the use of a suitable active-set estimate that enables us to update separately the variables estimated active and those estimated non-active at each iteration. Then, this approach is extended to solve minimization problems over the unit simplex and to the minimization of a function over the l1-ball. In the last chapter, a data filtering strategy for cluster analysis is proposed, based on solving a least squares problem with l0-norm regularization. Finally, for all the proposed methods, a theoretical analysis is carried out and numerical results are provided.