An Efficient Spectral Parameter for Accelerating the Conjugate Gradient Algorithm

Conjugate gradient (CG) methods are renowned for their efficiency in optimizing problems due to their speed and memory use. However, certain formulations of CG methods frequently change between different parameters, potentially omitting valuable values. Additionally, proving adequate descent properties in these methods often relies on a strategy that assumes the ability to exclude specific function values. This study introduces a novel spectral CG (SCG) algorithm that integrates both the spectral gradient parameter and conjugate gradient coefficient. This method stands out for its applicability to large-scale unconstrained optimization problems. Initial numerical findings employing a robust Wolfe line search are provided to showcase the method's efficiency. The spectral coefficient in the field of optimization is a concept used to enhance the performance of search algorithms in unconstrained optimization. It is typically employed in the context of line search methods to improve the convergence rate of algorithms. In optimization algorithms, the spectral coefficient helps adjust the step size to achieve faster convergence towards the optimal solution. The coefficient is calculated based on information from previous iterations of the algorithm. A common application of this concept is in the Spectral Gradient method, where the spectral coefficient is used to modify the gradient direction to accelerate the search process. In this method, the spectral coefficient is used to determine the step length in each iteration, aiming to achieve an optimal balance between speed and stability in finding the optimal solution