Optimization of unconstrained functions with sparse hessian matrices-newton-type methods

Newton-type methods for unconstrained optimization problems have been very successful when coupled with a modified Cholesky factorization to take into account the possible lack of positivedefiniteness in the Hessian matrix. In this paper we discuss the application of these methods to large problems that have a sparse Hessian matrix whose sparsity is known a priori. Quite often it is difficult, if not impossible, to obtain an analytic representation of the Hessian matrix. Determining the Hessian matrix by the standard method of finite-differences is costly in terms of gradient evaluations for large problems. Automat ic procedures that reduce the number of gradient evaluations by exploiting sparsity are examined and a new procedure is suggested. Once a sparse approximation to the Hessian matrix has been obtained, there still remains the problem of solving a sparse linear system of equations at each iteration, A modified Cholesky factorization can be used. However, many additional nonzeros (fill-in) may be created in the factors, and storage problems may arise. One way of approaching this problem is to ignore fill-in in a systematic manner. Such techniques are called partial factorization schemes. Various existing partial factorization are analyzed and three new ones are developed. The above algorithms were tested on a set of problems. The overall conclusions were that these methods perform well in practice.