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This paper explores the convergence ofnonlinear conjugate gradient methods without restarts, and with practical line searches. The analysis covers two classes ofmethods that are globally convergent on smooth, nonconvex functions. Some properties of the Fletcher-Reeves method play an important role in the first family, whereas the second family shares an important property with the Polak-Ribire method. Numerical experiments are presented.

. Much progress has been made in constrained nonlinear optimization in the past ten years, but most large-scale problems still represent a considerable obstacle. In this survey paper we will attempt to give an overview of the current approaches, including interior and exterior methods and algorithms based upon trust regions and line searches. In addition, the importance of software, numerical linear algebra and testing will be addressed. We will try to explain why the difficulties arise, how attempts are being made to overcome them and some of the problems that still remain. Although there will be some emphasis on the LANCELOT and CUTE projects, the intention is to give a broad picture of the state-of-the-art. 1 IBM T.J. Watson Research Center, P.O.Box 218, Yorktown Heights, NY 10598, USA 2 Parallel Algorithms Team, CERFACS, 42 Ave. G. Coriolis, 31057 Toulouse Cedex, France 3 Central Computing Department, Rutherford Appleton Laboratory, Chilton, Oxfordshire, OX11 0QX, England ...

Trust region subproblems arise within a class of unconstrained methods called trust region methods. The subproblems consist of minimizing a quadratic function subject to a norm constraint. This thesis is a survey of dierent methods developed to nd an approximate solution to the subproblem. We study the well-known method of More and Sorensen [18] and two recent methods for large sparse subproblems: the so-called Lanczos method of Gould et al. [7] and the Rendl and Wolkowicz algorithm [31]. The common ground to explore these methods will be semidenite programming. This approach has been used by Rendl and Wolkowicz [31] to explain their method and the More and Sorensen algorithm; we extend this work to the Lanczos method. The last chapter of this thesis is dedicated to some improvements done to the Rendl and Wolkowicz algorithm and to comparisons between the Lanczos method and the Rendl and Wolkowicz algorithm. In particular, we show some weakness of the Lanczos method and show that ...