Abstract
The study of efficient iterative algorithms for addressing nonlinear least-squares (NLS) problems is of great importance. The NLS problems, which belong to a special class of unconstrained optimization problems, are of particular interest because of the special structure of their gradients and Hessians. In this paper, based on the spectral parameters of Barzillai and Borwein (1998), we propose three structured spectral gradient algorithms for solving NLS problems. Each spectral parameter in the respective algorithms incorporates the structured gradient and the information gained from the structured Hessian approximation. Moreover, we develop a safeguarding technique for the first two structured spectral parameters to avoid negative curvature directions. Moreso, using a nonmonotone line-search strategy, we show that the proposed algorithms are globally convergent under some standard conditions. The comparative computational results on some standard test problems show that the proposed algorithms are efficient.
Highlights
Consider the general unconstrained optimization problem: min{f (x) ∶ x ∈ Rn}, (1.1)where f ∶ Rn → R is assumed to be twice continuously differentiable function and bounded below
Where f ∶ Rn → R is assumed to be twice continuously differentiable function and bounded below. Popular iterative algorithms, such as Newton’s algorithm and quasi-Newton algorithms, generate a sequence of iterates {xk} ⊂ Rn that eventually converges to some solutions of problem (1.1) using the following recurrence relation xk+1 = xk + αkdk, (1.2)
To avoid negative curvature directions, we provide a safeguarding technique for the first two of the proposed spectral parameters when they are nonpositive at a particular iteration
Summary
Where f ∶ Rn → R is assumed to be twice continuously differentiable function and bounded below. Barzilai and Borwein required Dk−1 to approximately satisfy the quasi-Newton equation (1.4) by finding αk ∈ R that minimizes the following least squares problem min α. Kobayashi et al [16] introduced a structured matrix-free algorithm that uses conjugate gradient direction to solve large–scale nonlinear least-squares problems. Their algorithm incorporated some approaches such as GN, LM, and SQN into the Dai and Liao conjugate gradient algorithm [17]. The two algorithms update their search directions by incorporating a structured vector, which approximately satisfies the structured secant equation, into the BB spectral parameters (1.7) and (1.8).
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