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.

本文提出了一种结构化对角Hessian逼近，用于解决非线性最小二乘（NLS）问题。我们设计了一个满足弱割线方程的改进结构矩阵。然后使用该结构化矩阵以类似于Andrei（2019）的论文的方式得出Hessian的结构化对角线近似值。通过解决最小化问题，我们通过最小化任意两个连续更新之间的偏差和更新后的对角矩阵的迹线，得出了Hessian的结构化对角线近似公式，从而满足了修改后的结构化弱割割方程。此外，我们在一些标准假设下显示了所提出算法的全局收敛性。在308个实例上对一些基准问题进行了比较数值实验，结果表明了该算法的有效性。最后，我们揭示了该算法在两个平面机器人运动控制中的适用性。