A Descent Generalized RMIL Spectral Gradient Algorithm for Optimization Problems

This study develops a new conjugate gradient (CG) search direction that incorporates a well defined spectral parameter while the step size is required to satisfy the famous strong Wolfe line search (SWP) strategy. The proposed spectral direction is derived based on a recent method available in the literature, and satisfies the sufficient descent condition irrespective of the line search strategy and without imposing any restrictions or conditions. The global convergence results of the new formula are established using the assumption that the gradient of the defined smooth function is Lipschitz continuous. To illustrate the computational efficiency of the new direction, the study presents two sets of experiments on a number of benchmark functions. The first experiment is performed by setting uniform SWP parameter values for all the algorithms considered for comparison. For the second experiment, the study evaluates the performance of all the algorithms by considering the exact SWP parameter values used for the numerical experiments as reported in each work. The idea of these experiments is to study the influence of parameters in the computational efficiency of various CG algorithms. The results obtained demonstrate the effect of the parameter value on the robustness of the algorithms.