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This work is concerned with the solution of the convex quadratic programming problem arising in training the learning machines named support vector machines. The problem is subject to box constraints and to a single linear equality constraint; it is dense and, for many practical applications, it becomes a large-scale problem. Thus, approaches based on explicit storage of the matrix of the quadratic form are not practicable. Here we present an easily parallelizable approach based on a decomposition technique that splits the problem into a sequence of smaller quadratic programming subproblems. These subproblems are solved by a variable projection method that is well suited to a parallel implementation and is very effective in the case of Gaussian support vector machines. Performance results are presented on well known large-scale test problems, in scalar and parallel environments. The numerical results show that the approach is comparable on scalar machines with a widely used technique and can achieve good efficiency and scalability on a multiprocessor system.