Preconditioned Krylov Subspace Methods in Nonlinear Optimization

One of the possible ways of solving general problems of constrained nonlinear optimization is to convert them into a sequence of unconstrained problems. Then the need arises to solve an unconstrained optimization problem reliably and efficiently. For this aim, Newton methods are usually applied, often in combination with sparse Cholesky decomposition. In practice, however, this approach may not be optimal in some cases and a suitable iterative algorithm for the solution of the Newton system may be preferred. In this paper, we show that for some type of problems the use of Krylov subspace methods with suitable preconditioners leads to considerable savings in computational time. Moreover, a new adaptive preconditioner based on the ellipsoid method is described in detail.