A family of the modified three-term Hestenes-Stiefel conjugate gradient method with sufficient descent and conjugacy conditions

To strengthen the three-term Hestenes-Stiefel conjugate gradient method proposed by Zhang et al., we suggest a modified version of it. For this purpose, by considering the Dai-Liao approach, the third term of Zhang et al. method is multiplied by a positive parameter which can be determined adaptively. To render an appropriate choice for the parameter of the search direction, we carry out a matrix analysis by which the sufficient descent property of the method is guaranteed. In the following, convergence analyses are discussed for convex and nonconvex cost functions. Eventually, numerical tests shed light on the efficiency of the performance of the proposed method.