Many problems in science and engineering can be formulated as nonlinear least-squares (NLS) problems. Thus, the need for efficient algorithms to solve these problems can not be overemphasized. In that sense, we introduce a generalized structured-based diagonal Hessian algorithm for solving NLS problems. The formulation associated with this algorithm is essentially a generalization of a similar result in Yahaya et al. (Journal of Computational and Applied Mathematics, pp. 113582, 2021). However, in this work, the structured diagonal Hessian update is derived under a weighted Frobenius norm; this allows other choices of the weighted matrix analogous to the Davidon-Fletcher-Powell (DFP) method. Moreover, to theoretically fill the gap in Yahaya et al. (Journal of Computational and Applied Mathematics, pp. 113582, 2021), we have shown that the proposed algorithm is R-linearly convergent under some standard conditions devoid of any safeguarding strategy. Furthermore, we experimentally tested the proposed scheme on some standard benchmark problems in the literature. Finally, we applied this algorithm to solve robotic motion control problem consisting of 3DOF (degrees of freedom).

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.