Implicitly restarted projection algorithm for solving optimization problems

An algorithm for solving the problem of minimizing a quadratic function subject to ellipsoidal constraints is introduced. This algorithm is based on the impHcitly restarted Lanczos method to construct a basis for the Krylov subspace in conjunction with a model trust region strategy to choose the step. The trial step is computed on the small dimensional subspace that lies inside the trust region.

One of the main advantages of this algorithm is the way that the Krylov subspace is terminated. We introduce a terminationcondition that allows the gradient to be decreased on that subspace.

A convergence theory for this algorithm is presented. It is shown that this algorithm is globally convergent and it shouldcope quite well with large scale minimization problems. This theory is sufficiently general that it holds for any algorithm that projects the problem on a lower dimensional subspace.