A Novel Generalized Conjugate Gradient Framework: Unlocking New Optimization Methods and Insights

Conjugate Gradient (CG) methods are efficient techniques in addressing large-scale, unconstrained optimization problems because of their efficiency and minimal storage requirement. This paper introduces a new family of generalized CG methods by linearly combining terms from the numerators and denominators of ten existing CG methods to propose a novel update parameter. We derive two new methods from this general framework and conduct a comprehensive numerical simulation against established methods. Numerical experiments on benchmark optimization problems show that one of the newly developed methods consistently surpasses conventional methods in both the speed of convergence and robustness. The second method competes favorably with existing methods. These findings indicate that the new class of generalized CG methods offers a promising direction for further research and practical applications in optimization techniques.