Nonlinear Convex Analysis and Optimization: An International Journal on Numerical, Computation and Applications

On the Class of Wei-Yao-Liu Conjugate Gradient Methods for Vector Optimization

Jamilu Yahaya, Poom Kumam, Sani Salisu, Auta Jonathan Timothy — Sun, 30 Jun 2024 00:00:00 +0000

Vector optimization problems (VOPs) are crucial research areas with widespread applications. The scalarization approach is commonly used to solve VOPs by transforming vector-valued functions into single-objective optimization. Despite its elegance, this method has the drawback of subjective weight selections. Alternatively, we propose five conjugate gradient (CG) methods designed for VOPs, where the set of Pareto-optimal points are obtained without weight selections, the methods are Wei-Yao-Liu (WYL) and four of its variants. Three of these methods lack sufficient descent conditions (SDC) in this context. However, we establish their global convergence using Wolfe line search. The remaining two methods fulfill SDC with the Wolfe line search, and their global convergence is further verified using the Wolfe line search. Importantly, our approach does not rely on regular restart or convexity assumptions associated with objective functions. We conduct numerical experiments to showcase the effectiveness of our methods, comparing them with the nonnegative PRP method. Through these experiments, we demonstrate the practical implementations of our proposed techniques.

A Descent Matrix-free Nonlinear Conjugate Gradient Algorithm for Impulse Noise Removal

Nasiru Salihu, Poom Kumam, Ibrahim Mohammed Sulaiman, Suraj Salihu — Sun, 30 Jun 2024 00:00:00 +0000

The convergence of the Polak, Ribie‘re-Polyak (PRP) conjugate gradient (CG) method requires some modifications for improved theoretical properties. In this article, we explore an optimal choice of the Perry conjugacy condition to propose a hybrid CG parameter for solving optimization and inverse problems, particularly in an image reconstruction model.
This parameter is selected to satisfy a combination of revised version of the PRP and Dai-Yuan (DY) CG methods. The numerical implementation includes inexact line search, showcasing the scheme's robustness (highest number of solved functions) compared to other known CG algorithms. The efficiency is shown in terms of Real error (RelErr), peak signal noise ratio (PNSR), and CPU time in seconds for impulse noise removal while for unconstrained minimization problems, the study evaluated the efficiency based on number of iterations, function evaluation, and CPU time in seconds. An interesting feature of the proposed method is its ability to converges to the minimizer regardless of the initial guess, relying on certain established assumptions.