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

Abstract

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.