Addendum to: A hybrid conjugate gradient method based on a quadratic relaxation of Dai–Yuan hybrid conjugate gradient parameter

Abstract

Here, necessary corrections on computing the hybridization parameter of the quadratic hybrid conjugate gradient method of Babaie-Kafaki [S. Babaie-Kafaki, A hybrid conjugate gradient method based on a quadratic relaxation of Dai-Yuan hybrid conjugate gradient parameter, Optimization, DOI: 10.1080/02331934.2011.611512, 2011] are stated in brief. Throughout, we use the same notations and equation numbers as in Babaie-Kafaki (2011).

Keywords:

• unconstrained optimization
• conjugate gradient method
• conjugacy condition

AMS Subject Classifications::

• 65K05
• 90C30
• 49M37

1. Two new quadratic hybrid conjugate gradient parameters

Recently, a continuous hybridization of the Dai–Yuan and Hestenes–Stiefel conjugate gradient (CG) parameters has been proposed in 1 based on a quadratic relaxation of a hybrid CG parameter proposed by Dai and Yuan 3. Although all the theoretical results of 1 are correct, its numerical results need some revisions due to an improper choice of the hybridization parameter.

It is remarkable that after some algebraic manipulations, it can be shown that the CG parameter β _k (θ _k ) defined by (2.5) with the hybridization parameter θ _k defined by (2.10) is equivalent to and consequently, in (2.14) we have . Thus, the approach of 1 for computing the hybridization parameter θ _k fails to generate a new CG method. This important defect of 1 demonstrates the necessity of proposing new hybridization parameters in order to generate new hybrid CG parameters in the quadratic form of (2.5). Here, two formulae for the hybridization parameter θ _k is suggested based on the extended conjugacy condition (t ≥ 0), as mentioned in Remark 2.1 of 1. More precisely, from (1.2) at the kth iteration the following quadratic equation is solved for finding the hybridization parameter θ _k :

which yields

Now, we adopt the approach of 2 in order to modify the CG parameter β _k (θ _k ) as follows:

and then, following the considerations of 2, we propose the CG parameters of our hybrid CG methods called and , respectively, computed based on the values of and , as follows:

Based on the numerical considerations of 2, the CG methods HQCG+ and HQCG– with t = 0.3 have been numerically compared with the CG methods hDY and hDYz. Figure 1(a) and (b) show the comparison results in the sense of the Dolan–Moré performance profile. As shown by the figure, HQCG+ is numerically efficient while HQCG– has a poor numerical performance. It can been shown that the CG methods HQCG+ and HQCG– have the important feature of global convergence for general functions.

Figure 1. Performance profiles for HQCG+, HQCG–, hDY and hDYz: (a) Effective function evaluations and (b) CPU time.

Acknowledgements

This research was in part supported by a grant from IPM (no. 90900023), and in part by the Research Council of Semnan University.