Gradient projection methods based on the Barzilai-Borwein spectral steplength choices are considered for quadratic programming problems with simple constraints. Well known nonmonotone spectral projected gradient methods and variable projection methods are discussed. For both approaches the behavior of different combinations of the two spectral steplengths is investigated. A nw adaptive stplength alternating rule is proposed, that becomes the basis for a generalized version of variable projection method (GVPM). Convergence results are given for the proposed approach and its effectiveness is shown by means of an extensive computational study on several test problems, including the special quadratic programs arising in training support vector machines. Finally, the GVPM behavior as inner QP solver in decomposition techniques for large-scale support vector machines is also evaluated.
Optimization Methods and Software 20 (2005), 353-378.