A note on the global convergence theorem of the scaled conjugate gradient algorithms proposed by Andrei

In (Andrei, Comput. Optim. Appl. 38:402-416, 2007), the efficient scaled conjugate gradient algorithm SCALCG is proposed for solving unconstrained optimization problems. However, due to a wrong inequality used in (Andrei, Comput. Optim. Appl. 38:402-416, 2007) to show the sufficient descent property for the search directions of SCALCG, the proof of Theorem 2, the global convergence theorem of SCALCG, is incorrect. Here, in order to complete the proof of Theorem 2 in (Andrei, Comput. Optim. Appl. 38:402-416, 2007), we show that the search directions of SCALCG satisfy the sufficient descent condition. It is remarkable that the convergence analyses in (Andrei, Optim. Methods Softw. 22:561-571, 2007; Eur. J. Oper. Res. 204:410-420, 2010) should be revised similarly.