Rabiu Bashir Yunus

This paper proposes a new spectral three‐term conjugate gradient (STTCG) method for solving large‐scale unconstrained optimization problems. The proposed method integrates the Liu–Storey conjugate gradient parameter with a modified three‐term correction and a safeguarded Barzilai–Borwein spectral scaling strategy to enhance both robustness and computational efficiency. The resulting search direction satisfies the sufficient descent condition independently of the line search procedure, while its conjugacy property is established under a positive curvature condition. Furthermore, the global convergence of the proposed method is proved under standard assumptions. Extensive numerical experiments on 120 benchmark test problems with dimensions ranging from n = 2 to n = 500, 000 demonstrate that STTCG is competitive with five state‐of‐the‐art conjugate gradient methods in terms of the number of iterations, function evaluations, and CPU time. To further demonstrate its practical applicability, the proposed method is employed in the trajectory tracking of a four‐degrees‐of‐freedom (4DOF) robotic manipulator, where it exhibits accurate tracking performance and favorable computational efficiency. MSC2020 Classfication: 65K05 | 90C52 | 90C56 | 94A08

This paper introduces a hybrid conjugate gradient (CG) method for unconstrained optimization with a spectral strategy, inspired by key advancements in existing CG techniques. The proposed method guarantees a descent direction at every iteration, regardless of the line search scheme employed. Its global convergence is rigorously established under the Wolfe line search conditions. Numerical experiments on benchmark optimization problems demonstrate that the proposed method outperforms the FR and RMIL methods across multiple performance metrics. Furthermore, its effectiveness is showcased in a neural network framework for predicting chickenpox and COVID-19 infection cases, highlighting its practical applicability in real-world scenarios.

Nonlinear least squares (NLS) models are extensively used as optimization frameworks in various scientific and engineering disciplines. This work proposes a novel structured conjugate gradient (SCG) method that incorporates a structured diagonal approximation for the second-order term of the Hessian, particularly designed for solving NLS problems. In addition, an acceleration scheme for the SCG method is proposed and analyzed. The global convergence properties of the proposed method are rigorously established under specific assumptions. Numerical experiments were conducted on large-scale NLS benchmark problems to evaluate the performance of the method. The outcome of these experiments indicates that the proposed method outperforms other approaches using the established performance metrics. Moreover, the developed approach is utilized to address the inverse kinematics challenge in controlling the motion of a robotic system with four degrees of freedom (4DOF).

Nonlinear least squares (NLS) problems are commonly encountered in a wide range of scientific and engineering fields, often demanding robust and efficient optimization methods for their solution. Traditional approaches for solving NLS problems frequently encounter difficulties related to high computational costs and memory usage, particularly in large-scale scenarios. This study introduces a novel structured Dai-Yuan two-term conjugate gradient (CG) method to address the minimization of NLS problems. By leveraging a Taylor series expansion of the objective function's Hessian, a structured vector approximation, representing a matrix-vector interaction, is constructed. This formulation adheres to a quasi-Newton condition. The method utilizes this structured approximation to enrich the conventional search direction with additional curvature information from the Hessian. The resulting search direction satisfies the required descent condition. Furthermore, numerical experiments on a range of benchmark problems demonstrate that the proposed method offers superior performance, outperforming several existing techniques.

Numerous scientific and technical domains have found use for nonlinear least squares (NLS). Conventional approaches to NLS problem solving often suffer from computational inefficiencies and high memory requirements, particularly when applied to large-scale systems. This paper presents the structured conjugate gradient coefficient using a structured secant-like approximation to solve NLS problems. The approach develops a structured vector approximation that captures the vector-matrix relationship using Taylor series expansions of the Hessian of the goal function. It is possible to incorporate more Hessian information into the traditional search direction, since this approximation satisfies a quasi-Newton condition. Furthermore, given conventional assumptions, a global convergence study is performed. Benchmark NLS tasks are used for numerical studies to assess the method's performance against alternative approaches. The results demonstrate the promise and success of the approach. Lastly, problems with image restoration are addressed using the suggested technique.

‎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.

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.