A Hybrid Three-Term Conjugate Gradient Algorithm for Solving Unconstrained Optimization Problems

Abstract

In this paper, we introduce a novel hybrid three-term conjugate gradient algorithm referred to as THREER, designed to address unconstrained optimization problems. The proposed approach integrates the -parameter introduced by Al-Neami with an additional third component derived from a rate-based vector ​, resulting in a search direction that preserves and enhances key characteristics of traditional conjugate gradient methods. A rigorous theoretical investigation establishes that the algorithm satisfies the sufficient descent condition regardless of the line search technique employed. Furthermore, the global convergence of the method is guaranteed under commonly accepted assumptions. Extensive numerical experiments conducted on large-scale benchmark problems reveal that THREER achieves superior performance when compared with several classical algorithms, particularly in terms of iteration count and function evaluations. These results highlight the algorithm’s robustness, efficiency, and potential for solving high-dimensional optimization tasks.