A diagonal Hestenes–Stiefel conjugate gradient algorithm with iterative complexity analysis and its application in robotic model

This paper presents a novel diagonal Hestenes–Stiefel conjugate gradient (DHS-CG) algorithm for solving large-scale unconstrained optimization problems. Building on the approaches presented in Dong et al. (2015) and Mohammad and Santos (2018), the proposed algorithm integrates a computationally efficient diagonal Hessian approximation into the HS-CG scheme. The algorithm constructs search directions that enjoy sufficient descent, trust region, and conjugacy properties without imposing restrictive line search conditions, thus ensuring efficiency robustness under Wolfe and Armijo-type line search strategies. We establish the global convergence and iterative complexity of the proposed algorithm under standard bounds and Lipschitz gradient assumptions. Extensive numerical experiments on benchmark problems, including large-scale cases with up to 500,000 variables, demonstrated that DHS-CG consistently outperformed state-of-the-art CG variants, such as CG_DESCENT and classical Hestenes–Stiefel (HS). In both low and high dimensional settings, the proposed algorithm achieves faster convergence, fewer function evaluations, and lower CPU time, highlighting its suitability for high performance scientific and engineering computations.