Development of a Spectral Conjugate Gradient Method for Solving Optimization Problems

Conjugate gradient methods have been favored to use for their efficiency in solving large-scale unconstrained optimization problems, primarily because of their low memory requirements and exclusive to use the first-order derivative information. In this paper, we introduce a spectral conjugate gradient method that enhances the classical approach by merge a spectral property directly into the determination of the search direction. At the core of our method lies a developed formulation of a spectral search direction and a more precisely adjusted conjugate gradient coefficient, both derived as extensions of established conjugacy condition. To ensure numerical stability, we also include a correction term that accounts for the limitations of machine precision. Our theoretical analysis confirms that the developed method generates search directions satisfying the descent condition, which is critical for ensuring convergence. To assess its real-world effectiveness, we subjected the spectral conjugate gradient method to an extensive set of numerical experiments and benchmarked its performance against that of a standard conjugate gradient method. By using range of test problems, our method consistently delivered superior results, particularly in reducing the number of function evaluations and exhibiting improved scalability in higher-dimensional settings. These findings strongly indicate the spectral conjugate gradient method’s potential as a reliable and efficient tool for optimization. Future research may explore further refinements to the method’s theoretical foundations, investigate its performance in constrained or stochastic environments, and apply it to practical optimization challenges such as neural network training, signal recovery, structural design, and control system calibration.