Combining search directions using gradient flows

 The efficient combination of directions is a significant problem in line search methods that either use negative curvature, or wish to include additional information such as the gradient or different approximations to the Newton direction.

In this paper we describe a new procedure to combine several of these directions within an interior-point primal-dual algorithm. Basically, we combine in an efficient manner a modified Newton direction with the gradient of a merit function and a direction of negative curvature, if it exists. We also show that the procedure is well-defined, and it has reasonable theoretical properties regarding the rate of convergence of the method.

We also present numerical results from an implementation of the proposed algorithm on a set of small test problems from the CUTE collection.

Authors and Affiliations

1. School of Engineering, Univ. Rey Juan Carlos, Madrid, Spain, e-mail: j.moguerza@escet.urjc.es. Research supported by Spanish MEC grant TIC2000-1750-C06-04 and CAM project 07T/0005/2001, , , , , , ES

   Javier M. Moguerza

2. Dept. of Statistics and Econometrics, Univ. Carlos III de Madrid, Spain, e-mail: fjp@estecon.uc3m.es. Research supported by Spanish MEC grants BEC2000-0167 and PB98-0728, , , , , , ES

   Francisco J. Prieto

Received: November 2000 / Accepted: October 2002 Published online: February 14, 2003

Key Words. negative curvature – primal-dual methods – interior-point methods – nonconvex optimization – line searches

Mathematics Subject Classification (1991): 49M37, 65K05, 90C30

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