Different Types of Three-Term CG-Methods with Sufficient Descent and Conjugacy Conditions

1 Abbas Y. Al-Bayati, 2 Hawraz N. Al-Khayat
1 College of Telafer Basic Education, Mathematics, Mosul University, 2 College of Computer Sciences and Mathematics, Mathematics, Mosul University

It is very important to generate a descent search direction independent of line searches in
showing the global convergence of conjugate gradient methods. Recently, Zhang et al. proposed
a three-term of PR method (TTPR) and HS method (TTHS), both of which can produce
sufficient descent condition. In this paper, we treat two subjects: we first consider new unified
formula of three-term CG algorithm, second we suggested new scaled three-term algorithm
based on Birgin-Martínez algorithm and which satisfied both the descent and conjugacy
conditions are proposed. This algorithms are modification of the Hestenes-Stiefel and Birgin-
Martínez algorithms, also the algorithms could be considered as a modification of the
memoryless BFGS quasi-Newton method. Our algorithms can proved the global convergence
property and more efficiently than HS and BM algorithms in numerical results.

Keywords: Three-Term Conjugate Gradient, Global Convergence, Unconstrained Optimization,
Descent Direction, Conjugacy Condition, Memoryless BFGS.


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