Alternative structured spectral gradient algorithms for solving nonlinear least-squares problems

Abstract

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New CG-Based Algorithms With Second-Order Curvature Information for NLS Problems and a 4DOF Arm Robot Model

“…If ∥gk ∥ were bounded away from 0, the equation ( 56) would not hold since Pk+1 ≤ k + 2 by (47). Hence, if ηmax < 1, then by (47),…”

“…=J (x) T J (x) + P(x), (14) where J (x) = R ′ (x) represents the Jacobian matrix of the residual function, and the matrix P(x) corresponds to the second term mentioned in equation (14). First, we will state as shown in [31], [32], [47] e.t.c. The structured vector approximation, which is an action of a vector on a matrix, is derived from Taylor series approximations of the Hessian of the objective function defined in (14) such that the following secant equation is fulfilled:…”

A Modified Structured Spectral HS Method for Nonlinear Least Squares Problems and Applications in Robot Arm Control

“…To overcome some of the problems they encountered in [22], Yahaya et al [23] developed structured, quasi-Newton-based methods with the second term being approximated using higher-order Taylor's series expansion. In subsequent efforts, Muhammad and Waziri [24] suggested a couple of BB-like algorithms for handling NLS problems and, later, Yahaya et al [25] proposed a modified version by improving the numerical performance of [24]. However, the two algorithms demand several safeguarding techniques, especially when the spectral parameter is non-positive at some iteration points.…”

“…This is achieved by incorporating modified secant updates in the second part of the Hessian matrix. The motivation behind this work was the expectation that our matrix-free approach, with the inclusion of the spectral parameter update, could capture more information from the objective function and eliminate the need for a safeguarding strategy, unlike the method proposed in [25].Below is an overview of the key contributions of this study:…”

Multi-order analytical solving computation of rainstorm causal decomposition during typhoons using a designed key–lock quasi-Newton optimizing derivation

“…Newton's method of nonlinear programming is a conventional identification algorithm that uses the initial predicted parameter sequence with the sensitivity coefficient (i.e., Jacobian) matrix to search for the root (optimal solution ) of the first-order partial derivative of the objective function . To improve the convergence speed of parameter identification, the Gauss–Newton method uses the second-order partial derivative composed of a square matrix of the objective function versus parameters (Hessian matrix) to describe the local curvature of parameter function and optimize parameters along a shorter and more direct path than Newton's method [ 36 ]. However, calculating the inverse of a high-dimensional Hessian matrix is an expensive operation, which can be solved using various decomposition or approximate iterative methods.…”