A Note on the Descent Property Theorem for the Hybrid Conjugate Gradient Algorithm CCOMB Proposed by Andrei

In [1] (Hybrid Conjugate Gradient Algorithm for Unconstrained Optimization J. Optimization. Theory Appl. 141 (2009) 249 - 264), an efficient hybrid conjugate gradient algorithm, the CCOMB algorithm is proposed for solving unconstrained optimization problems. However, the proof of Theorem 2.1 in [1] is incorrect due to an erroneous inequality which used to indicate the descent property for the search direction of the CCOMB algorithm. It is also remarkable that the proof of the Theorem 2.2 should be revised. Following the notations in [1], the main goal of this note is to provide some necessary corrections to rectify the mentioned issues.