On optimality of two adaptive choices for the parameter of Dai-Liao method

Orthonormal matrices are a class of well-conditioned matrices with the least spectral condition number. Here, at first it is shown that a recently proposed choice for parameter of the Dai-Liao nonlinear conjugate gradient method makes the search direction matrix as close as possible to an orthonormal matrix in the Frobenius norm. Then, conducting a brief singular value analysis, it is shown that another recently proposed choice for the Dai-Liao parameter improves spectral condition number of the search direction matrix. Thus, theoretical justifications of the two choices for the Dai-Liao parameter are enhanced. Finally, some comparative numerical results are reported.