Appl Math Optim 43:117–128 (2001)
2001 Springer-Verlag New York Inc.
A Spectral Conjugate Gradient Method
for Unconstrained Optimization
E. G. Birgin
and J. M. Mart´ınez
Department of Computer Science,
IME-USP, University of S˜ao Paulo,
Rua do Mat˜ao, 1010 - Cidade Universit´aria,
05508-900 S˜ao Paulo SP, Brazil
Department of Applied Mathematics,
IMECC-UNICAMP, University of Campinas,
CP 6065, 13081-970 Campinas SP, Brazil
Abstract. A family of scaled conjugate gradient algorithms for large-scale uncon-
strained minimization is deﬁned. The Perry, the Polak–Ribi`ere and the Fletcher–
Reeves formulae are compared using a spectral scaling derived from Raydan’s
spectral gradient optimization method. The best combination of formula, scaling
and initial choice of step-length is compared against well known algorithms using
a classical set of problems. An additional comparison involving an ill-conditioned
estimation problem in Optics is presented.
Key Words. Unconstrained minimization, Spectral gradient method, Conjugate
AMS Classiﬁcation. 49M07, 49M10, 90C06, 65K.
The ﬁrst author was supported by FAPESP (Grants 98/07704-1 and 99/08029-9) and PRONEX-
Optimization 76.79.1008-00. The second author was supported by FAPESP (Grant 90/3724-6), CNPq, FAEP-
UNICAMP and PRONEX-Optimization 76.79.1008-00.