ABSTRACT Recently, several modifications to the Rivaie, Mustafa, Ismail, and Leong (RMIL) conjugate gradient (CG) method have been proposed. However, these modifications do not fully capture the true nature of the parameter or account for the extended conjugacy condition. In this paper, we introduce a revised RMIL (rRMIL) CG parameter, along with three spectral CG (SCG) parameters, to solve unconstrained nonlinear CG problems and a real‐world model, including portfolio selection. To eliminate the need for an exact line search, one of the SCG parameters generalizes the others by utilizing an extended conjugacy condition. The first search direction in the SCG method is guaranteed to satisfy the well‐known sufficient descent condition (SDC) using the geometric properties of the search directions and gradients (via the cosine formula for the angles between vectors), relying instead on the pure conjugacy condition. This, together with the other two parameters, demonstrates their SDC without the need for a line search. The global convergence of the proposed SCG methods is established under mild assumptions. Numerical experiments on standard nonlinear problems and portfolio selection models demonstrate that the generalized SCG method is both promising and efficient.
Salihu et al. (Sun,) studied this question.