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


Department of Applied Mathematics, School of Mathematics and Computer Science, Damghan University, Damghan, Iran.


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.


1. N. Andrei, Hybrid conjugate gradient algorithm for unconstrained optimization, J. Optim. Theory Appl 141(2), 249-264 (2009).

2. Y.H. Dai, Y. Yuan, A nonlinear conjugate gradient method with a strong global convergenc eproperty, SIAM Journal Optimization, 10(21), 348-358 (1991).

3. Y.H. Dai, J.Y. Han, G.H. Liu, and D.F. Sun, Convergence properties of nonlinear conjugate gradient methods, SIAM Journal Optimization, 10(21), 348-358 (1991).

4. J.C. Gilbert, J. Nocedal, Global convergence properties of conjugate gradient methods for optimization, SIAM Journal Optimization, 2(1), 21-42 (1992).

5. R. Polak, G. Ribiere, Note sur la convergence de mthodes de directions conjugues, Rev. Fr. Inf. Rech. Oper. 3e Annee, 3(1), 35-43 (1969).

6. P. Wolfe, Convergence conditions for ascent methods, SIAM Rev., 11(2), 226{235 (1969).

7. P. Wolfe, Convergence conditions for ascent methods, II: Some corrections, SIAM Rev., 13(2), 185-188 (1971).