Abstract
This paper suggests a new limited memory trust region algorithm for large unconstrained black box least squares problems, called LMLS. Main features of LMLS are a new non-monotone technique, a new adaptive radius strategy, a new Broyden-like algorithm based on the previous good points, and a heuristic estimation for the Jacobian matrix in a subspace with random basis indices. Our numerical results show that LMLS is robust and efficient, especially in comparison with solvers using traditional limited memory and standard quasi-Newton approximations.
Highlights
In this paper, we consider the unconstrained nonlinear least squares problem min f (x ):= E(x) 2 s.t. x ∈ Rn, (1)with high-dimensional x ∈ Rn and continuously differentiable E : Rn → Rr (r ≥ n), possibly expensive
To solve the least squares problem (1), trust region methods use linear approximations of the residual vectors to make surrogate quadratic models whose accuracy are increased by restricting their feasible points
We describe all steps of a new limited memory algorithm, called LMLS using the new subspace direction (7), the new non-monotone technique (10), the new adaptive radius strategy (11), and BroydenLike
Summary
There has been a huge amount of literature on least squares and its applications. There are many trust region methods using the finite difference method for the Jacobian matrix estimation such as CoDoSol and STRSCNE by Bellavia et al (2004, 2012), NMPNTR by Kimiaei (2016), NATRN and NATRLS by Amini et al (2016); Amini et al (2016), LSQNONLIN from the MATLAB Toolbox, NLSQERR (an adaptive trust region strategy) by Deuflhard (2011), and DOGLEG by Nielsen (2012) They are suitable for small- and medium-scale problems. To solve the least squares problem (1), trust region methods use linear approximations of the residual vectors to make surrogate quadratic models whose accuracy are increased by restricting their feasible points These methods use a computational measure to identify whether an agreement between an actual reduction of the objective function and a predicated reduction of surrogate quadratic model function is good or not. Trust region radii are reduced possibly many times, leading to the production of quite a small radius, or even a failure
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