A New Efficient Conjugate Gradient Scheme for Solving Unconstrained Nonlinear Problems

Abstract

This paper introduces a new hybrid Conjugate Gradient (CG) algorithm, referred to as KBH, for large-scale unconstrainedoptimization. The proposed method constructs the search direction using an adaptively weighted convex combination of the conjugacy parameters from Conjugate Descent (CD), Polak--Ribiere--Polyak (PRP), and Fletcher--Reeves (FR) method. This formulation seeksto merge the reliable descent characteristics of CD and FR with the strong practical efficiency often observed for PRP, while maintaining numerical stability. The adaptive weights are obtained analytically from the classical conjugacy condition, yielding a true three-term hybrid approach with data-dependent coefficients. Assuming standard smoothness conditions and employing a strong Wolfe line search, the sufficient descent property of the resulting directions is shown, and global convergence of the method is established. Moreover, safeguarding, truncation, and restart strategies are added to enhance robustness in practical use. Comprehensive numerical tests on a wide set of large-scale benchmark problems show that KBH achieves competitive results relative to several classical CG methods in terms of iteration numbers, function evaluations, and CPU time.