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This article presents a new hybrid conjugate gradient (CG) algorithm for solving unconstrained optimization problem. The search direction is defined as a combination of Hestenes–Stiefel (HS) and the Liu–Storey (LS) CG parameters and is close to the direction of the memoryless Broyden–Fletcher–Goldferb–Shanno (BFGS) quasi-Newton direction. In addition, the search direction is descent and bounded. The global convergence of the algorithm is obtained under the Wolfe-type and Armijo-type line searches. Numerical experiments on some benchmark test problems is carried out to depict the efficiency and robustness of the hybrid algorithm. Furthermore, a practical application of the algorithm in motion control of robot manipulator and image restoration is provided. [ABSTRACT FROM AUTHOR]

Conjugate gradient methods have played a vital role in finding the minimizers of large-scale unconstrained optimization problems due to the simplicity of their iteration, convergence properties and their low memory requirements. Based on the Liu-Storey conjugate gradient method, in this paper, we present a Liu-Storey type method for finding the minimizers of large-scale unconstrained optimization problems. The direction of the proposed method is constructed in such a way that the sufficient descent condition is satisfied. Furthermore, we establish the global convergence result of the method under the standard Wolfe and Armijo-like line searches. Numerical findings indicate that our presented approach is efficient and robust in solving large-scale test problems. In addition, an application of the method is explored. [ABSTRACT FROM AUTHOR]

This article proposes a structured diagonal Hessian approximation for solving non-linear least-squares (NLS) problems. We devised a modified structured matrix that satisfies the weak secant equation. This structured matrix is then used to derive the structured diagonal approximation of the Hessian in a similar pattern as the paper of Andrei (2019). By solving a minimization problem, we derived the formulation of the structured diagonal approximation of the Hessian by minimizing the deviation between any two successive updates and the trace of the updated diagonal matrix so that the modified structured weak secant equation is satisfied. More so, we show the global convergence of the proposed algorithm under some standard assumptions. Comparative numerical experiments on some benchmark problems with 308 instances show the efficacy of the proposed algorithm. Finally, we reveal the applicability of the proposed algorithm in the motion control of two planar robots. [ABSTRACT FROM AUTHOR]