this paper has appeared in

this article I will attempt to review the most recent advances in the theory of unconstrained optimization, and will also describe some important open questions. Before doing so, I should point out that the value of the theory of optimization is not limited to its capacity for explaining the behavior of the most widely used techniques. The question

Abstract. Computational experience with several limited-memory quasi-Newton and truncated Newton methods for unconstrained nonlinear optimization is described. Comparative tests were conducted on a well-known test library [J. J. Mor, B. S. Garbow, and K. E. Hillstrom, ACM Trans. Math. Software, 7 (1981), pp. 17-41], on several synthetic problems allowing control of the clustering of eigenvalues in the Hessian spectrum, and on some large-scale problems in oceanography and meteorology. The results indicate that among the tested limited-memory quasi-Newton methods, the L-BFGS method [D. C. Liu and J. Nocedal, Math. Programming, 45 (1989), pp. 503-528] has the best overall performance for the problems examined. The numerical performance of two truncated Newton methods, differing in the inner-loop solution for the search vector, is competitive with that of L-BFGS. Key words, limited-memory quasi-Newton methods, truncated Newton methods, synthetic cluster functions, large-scale unconstrained minimization AMS subject classifications. 90C30, 93C20, 93C75, 65K10, 76C20 1. Introduction. Limited-memory quasi-Newton (LMQN) and truncated Newton

Abstract. This paper reviews the development of different versions of nonlinear conjugate gradient methods, with special attention given to global convergence properties.

In this paper we propose a new line search algorithm that ensures global convergence of the PolakRibi `ere conjugate gradient method for the unconstrained minimization of nonconvex differentiable functions. In particular, we show that every limit point produced by the Polak-Ribi`ere iteration is a stationary point of the objective function. Moreover, we prove that, asymptotically, the first stationary point along the search direction can be accepted and that, under strong convexity assumptions, the known global convergence results can be reobtained as a special case. From a computational point of view, we may expect that an algorithm incorporating the stepsize acceptance rules proposed here will retain the same good features of the Polak-Ribi`ere method, while avoiding pathological situations. 1 Introduction The objective of this paper is that of defining a new globally convergent implementation of the PolakRibi `ere conjugate gradient method for the unconstrained minimization of a ...

An investigation of incremental 4D-Var using