(untitled) (European Journal of Pure and Applied Mathematics, 2025)
In this century, the application of optimization methods is frequently utilized in numerous fields like image restoration, electrical engineering, medical science, machine learning (ML), signal processing as well as many others. In this paper, we choose to improve and maintain one of the most popular, low-memory, and simple algorithms of optimization methods. This method is known as the conjugate gradient method (CGM). Here, we develop a new 3-term CGM with several search directions in the third term suitable for any CGM related to the Fletcher-Reeves method. Apart from that, we propose a new 3-term CGM with mild conditions for any method in relation to the Polak-Ribière-Polyak method. The proposed method satisfies the descent and convergence properties. Moreover, in the numerical findings section, we perform a comparison of the new method with several renowned methods that have emerged in this century, such as CG-Descent 6.8 and nonnegative Dai-Liao methods utilizing more than 180CUTEst library functions. The numerical findings indicate that the novel approach surpasses recent methodologies. These numerical findings encompass the count of gradient assessments, function assessments, CPU duration as well as iteration count. Additionally, we discussed the implementation of the CG method in image restoration and pi-electric circuits.