This paper proposes a modification to the Hestenes-Stiefel (HS) method by devising a spectral parameter using a modified secant relation to solve nonlinear least-squares problems. Notably, in the implementation, the proposed method differs from existing approaches, in that it does not require a safeguarding strategy and its Hessian matrix is positive and definite throughout the iteration process. Numerical experiments are conducted on a range of test problems, with 120 instances to demonstrate the efficacy of the proposed algorithm by comparing it with existing techniques in the literature. However, the results obtained validate the effectiveness of the proposed method in terms of the standard metrics of comparison. Additionally, the proposed approach is applied to address motion-control problems in a robotic model, resulting in favorable outcomes in terms of the robot’s motion characteristics.

‎Nonlinear least-squares (NLS) problems find extensive applications across various fields within the applied sciences‎. ‎Conventional methods for solving NLS problems often face challenges related to computational efficiency and memory requirements‎, ‎especially when dealing with large-scale systems‎. ‎In this paper‎, ‎the solution to the minimization of nonlinear least squares problems has been obtained using a proposed structured accelerated three-term conjugate gradient method‎, ‎in which from Taylor series approximations of the objective function's Hessian‎, ‎the structured vector approximation involving a vector's action on a matrix is obtained‎. ‎This ensures the satisfaction of a quasi-Newton condition‎. ‎The technique then employs the structured vector approximation to incorporate additional information from the Hessian of the goal function into the standardized search direction‎. ‎The proposed method's search direction fulfills the necessary descent criterion‎. ‎Additionally‎, ‎numerical tests performed on various test problems show that the suggested approach is remarkably efficient‎, ‎surpassing some existing competitors‎.

This article develops two new algorithms of three-term conjugate gradient (TTCG) coefficients to handle nonlinear least-squares (NLS) problems using the structured secant equation. Motivated by improved conjugacy and sufficient descent conditions, we developed two new formulations of CG coefficients. Furthermore, with these parameters, two search directions are proposed that satisfy the sufficient descent condition, which further enhances the efficiency of the proposed strategies. One key advantage of the proposed techniques is their low memory requirements, rendering them suitable for large-scale nonlinear least squares problems. Moreover, some mild suppositions and a non-monotone line search are used to establish the global convergence properties of the methods. More so, we investigate the robustness and effectiveness of the proposed methods numerically by performing experiments on benchmark test problems, and their performance is compared against existing methods. The outcomes of these experiments indicate that the proposed methods outperform the other techniques regarding the metrics of comparison adopted. Finally, the algorithms are applied to an extension of the model of robotic motion control of four degrees of freedom (4DOF), resulting in positive outcomes for the robot’s motion traits.