Abstract
Many problems in science and engineering can be formulated as nonlinear least-squares (NLS) problems. Thus, the need for efficient algorithms to solve these problems can not be overemphasized. In that sense, we introduce a generalized structured-based diagonal Hessian algorithm for solving NLS problems. The formulation associated with this algorithm is essentially a generalization of a similar result in Yahaya <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> (Journal of Computational and Applied Mathematics, pp. 113582, 2021). However, in this work, the structured diagonal Hessian update is derived under a weighted Frobenius norm; this allows other choices of the weighted matrix analogous to the Davidon-Fletcher-Powell (DFP) method. Moreover, to theoretically fill the gap in Yahaya <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> (Journal of Computational and Applied Mathematics, pp. 113582, 2021), we have shown that the proposed algorithm is R-linearly convergent under some standard conditions devoid of any safeguarding strategy. Furthermore, we experimentally tested the proposed scheme on some standard benchmark problems in the literature. Finally, we applied this algorithm to solve robotic motion control problem consisting of 3DOF (degrees of freedom).
Highlights
I N this research article, we propose generalized structuredbased quasi-Newton algorithm for nonlinear least-squares problems of the following form: min f (x), x∈Rn 1 f (x) = m (ri(x))2 = 1 ∥r(x)∥2, 2 (1)i=1 where the residual, ri : Rn → R is a smooth function for each i = 1, 2, · · ·, m
We have proposed an algorithm for computing a minimizer of nonlinear least-squares problems
The developed algorithm is essentially based on a standard quasi-Newton class of algorithms; we called the algorithm ’generalized structured based diagonal algorithm’ (GSDA)
Summary
I N this research article, we propose generalized structuredbased quasi-Newton algorithm for nonlinear least-squares problems of the following form: min f (x), x∈Rn 1 f (x) = m (ri(x))2 = 1 ∥r(x)∥2, 2 (1). I=1 where the residual, ri : Rn → R is a smooth function for each i = 1, 2, · · · , m. We assume that for higherdimensional problems, i.e., when (n is large), the Jacobian matrix of r, J(x)T is not stored explicitly; we can evaluate the matrix-vector product say, JT v, where v ∈ Rm. the gradient, g(x) and Hessian, H(x) of f are defined as follows:. M g(x) = ri(x)∇ri(x) = J(x)T r(x), (2) i=1 and m m
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
