In this article, two Hager-Zhang (HZ) type projection algorithms are presented for large-dimension nonlinear monotone problems and sparse signal recovery in compressed sensing. This goal is attained ...by conducting singular value analysis of a nonsingular HZ-type search direction matrix as well as applying the idea by Piazza and Politi J Comput Appl Math. 2002;143(1):141-144 and minimizing the Frobenius norm of an orthornormal matrix. The paper attempts to fill the gap in the work of Hager and Zhang Pac J Optim. 2006;2(1):35-58, Waziri et al. Appl Math Comput. 2019;361:645-660, Sabi'u et al. Appl Numer Math. 2020;153:217-233 and Babaie-Kafaki 4OR-Q J Oper Res. 2014;12:285-292, where the sufficient descent or global convergence condition is not satisfied when the HZ parameter is in the interval
$ (0,\frac {1}{4}) $
(
0
,
1
4
)
. The proposed schemes are also suitable for solving non-smooth nonlinear problems. Also, by employing some mild conditions, global convergence of the schemes are established, while numerical comparison with four effective HZ-type methods show that the new methods are efficient. Furthermore, to illustrate their practical application, both methods are applied to solve the
$ \ell _1 $
ℓ
1
-norm regularization problems to recover a sparse signal in compressive sensing. The experiments conducted in that regard show that the methods are promising and perform better than two other methods in the literature.
We proposed a matrix-free direction with an inexact line search technique to solve system of nonlinear equations by using double direction approach. In this article, we approximated the Jacobian ...matrix by appropriately constructed matrix-free method via acceleration parameter. The global convergence of our method is established under mild conditions. Numerical comparisons reported in this paper are based on a set of large-scale test problems and show that the proposed method is efficient for large-scale problems.
In this paper, we present two choices of structured spectral gradient methods for solving nonlinear least squares problems. In the proposed methods, the scalar multiple of identity approximation of ...the Hessian inverse is obtained by imposing the structured quasi-Newton condition. Moreover, we propose a simple strategy for choosing the structured scalar in the case of negative curvature direction. Using the nonmonotone line search with the quadratic interpolation backtracking technique, we prove that these proposed methods are globally convergent under suitable conditions. Numerical experiment shows that the methods are competitive with some recently developed methods.
Two new conjugate residual algorithms are presented and analyzed in this article. Specifically, the main functions in the system considered are continuous and monotone. The methods are adaptations of ...the scheme presented by Narushima et al. (SIAM J Optim 21: 212–230, 2011). By employing the famous conjugacy condition of Dai and Liao (Appl Math Optim 43(1): 87–101, 2001), two different search directions are obtained and combined with the projection technique. Apart from being suitable for solving smooth monotone nonlinear problems, the schemes are also ideal for non-smooth nonlinear problems. By employing basic conditions, global convergence of the schemes is established. Report of numerical experiments indicates that the methods are promising.
Vast applications of derivative-free methods to restore the blurred images in compressive sensing has become an important trend in recent years. This research, a double direction method for better ...image restoration is proposed. Besides this, two double direction algorithms to solve constrained monotone nonlinear equations are presented. The main idea employed in the first algorithm is to approximate the Jacobian matrix via acceleration parameter to propose an effective derivative-free method. The second algorithm involve hybridizing the scheme of the first algorithm with Picard–Mann hybrid iterative scheme. In addition, the step length is calculated using inexact line search technique. The proposed methods are proven to be globally convergent under some mild conditions . The numerical experiment, shown in this paper, depicts the efficiency of the proposed methods. Furthermore, the second method is successfully applied to handle the
ℓ
1
-norm regularization problem in image recovery which exhibits a better result than the existing method in the previous literature.
The classical Dai–Kou scheme (SIAM J Optim 23(1):296–320, 2013), which is popularly referred to as CGOPT, represents one of the most numerically efficient methods for unconstrained optimization. This ...article exploits nice properties of the scheme together with the projection method to present an adaptive Dai–Kou type method for solving constrained system of nonlinear monotone equations. The new scheme utilizes the backtracking line search strategy by Zhang and Zhou (J Comput Appl Math 196:478–484, 2006) to determine the step-size in the algorithm. In addition, the scheme requires less memory to implement as it avoids computing Jacobian matrix. This attribute makes it an ideal choice for large-scale problems. Other attributes of the scheme include its ability to generate search directions that satisfy the vital condition for global convergence and its applications to signal and image reconstruction problems in compressive sensing. The global convergence of the new scheme is proved using basic assumptions and numerical experiments conducted suggests that the new approach has a clear edge over four iterative schemes in the literature.
In this paper, a single direction with double step length method for solving systems of nonlinear equations is presented. Main idea used in the algorithm is to approximate the Jacobian via ...acceleration parameter. Furthermore, the two step lengths are calculated using inexact line search procedure. This method is matrix-free, and so is advantageous when solving large-scale problems. The proposed method is proven to be globally convergent under appropriate conditions. The preliminary numerical results reported in this paper using a large-scale benchmark test problems show that the proposed method is practically quite effective.
This work proposes a structured diagonal Gauss–Newton algorithm for solving zero residue nonlinear least-squares problems. The matrix corresponding to the Gauss–Newton direction is approximated with ...a diagonal matrix that satisfies the structured secant condition. Using a derivative-free Armijo-type line search with some appropriate conditions, we prove that the proposed algorithm converges globally. Furthermore, the algorithm achieved R-linear convergence rate for zero residue problems. Numerical result shows that the algorithm is competitive with the existing algorithms in the literature.
We propose two positive parameters based on the choice of Birgin and Martínez search direction. Using the two classical choices of the Barzilai‐Borwein parameters, two positive parameters were ...derived by minimizing the distance between the relative matrix corresponding to the propose search direction and the scaled memory‐less Broyden–Fletcher–Goldfarb‐Shanno (BFGS) matrix in the Frobenius norm. Moreover, the resulting direction is descent independent of any line search condition. We established the global convergence of the proposed algorithm under some appropriate assumptions. In addition, numerical experiments on some benchmark test problems are reported in order to show the efficiency of the proposed algorithm.
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