A derivative-free three-term Hestenes–Stiefel type method for constrained nonlinear equations and image restoration

Ibrahim, Abdulkarim Hassan; Hassan, Basim A.; Abubakar, Auwal Bala; Abubakar, Jamilu

doi:10.1080/00207160.2021.1946043

Cited by 16 publications

(10 citation statements)

References 47 publications

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“…Considering the simplicity and low storage requirement of the conjugate gradient method [20,21], several researchers combined the projection technique of Solodov and Svaiter [56] with the conjugate gradient methods to solve large-scale nonlinear equations, see [38,13,42,40,43,1,7,45,41,12,47,10,9,46,39,52,2,3,8,53,11,37,44,4,6,36] and references therein. Based on the projection method, Gao and He [35] introduced an efficient three-term derivative-free method for solving nonlinear monotone equations with convex constraints (1) by choosing a part of the Liu-Storey (LS) conjugate parameter as a new conjugate parameter.…”

Section: Introductionmentioning

confidence: 99%

Projection method with inertial step for nonlinear equations: Application to signal recovery

Ibrahim

Sun

Chaipunya

et al. 2023

JIMO

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<p style='text-indent:20px;'>In this paper, using the concept of inertial extrapolation, we introduce a globally convergent inertial extrapolation method for solving nonlinear equations with convex constraints for which the underlying mapping is monotone and Lipschitz continuous. The method can be viewed as a combination of the efficient three-term derivative-free method of Gao and He [Calcolo. 55(4), 1-17, 2018] with the inertial extrapolation step. Moreover, the algorithm is designed such that at every iteration, the method is free from derivative evaluations. Under standard assumptions, we establish the global convergence results for the proposed method. Numerical implementations illustrate the performance and advantage of this new method. Moreover, we also extend this method to solve the LASSO problems to decode a sparse signal in compressive sensing. Performance comparisons illustrate the effectiveness and competitiveness of our algorithm.</p>

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Section: Introductionmentioning

confidence: 99%

Projection method with inertial step for nonlinear equations: Application to signal recovery

Ibrahim

Sun

Chaipunya

et al. 2023

JIMO

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“…Combining this with the second inequality in (23), we have {d † k } is bounded, and it further implies…”

Section: |E(x) − E(y)|| ≤ L||x − Y||;mentioning

confidence: 74%

“…Remark The Assumption (A3) was originally introduced by Solodov and Svaiter 37 to establish the global convergence of projection method for variational inequality problem. Later on, it was used by References 23,38,39 to prove the global convergence of the projection methods for constrained nonlinear equations. It should be noted that Assumption (A3) is satisfied if the mapping

E

is monotone, that is,

{\left[E(x)-E(y)\right]}^{\top}\left(x-y\right)\ge 0,\kern0.3em \forall \kern0.3em x,\kern0.3em y\in {\mathbb{R}}^n,\kern1.00em

or pseudo‐monotone, that is,

E{(y)}^{\top}\left(x-y\right)\ge 0\Rightarrow E{(x)}^{\top}\left(x-y\right)\ge 0,\kern0.3em \forall \kern0.3em x,\kern0.3em y\in {\mathbb{R}}^n,

but not vice versa (see Reference 40 for more details).…”

Section: Global Convergence Analysismentioning

confidence: 99%

“…Therefore, this work will only focus on the DFP methods, such as spectral gradient projection methods 7,8 and conjugate gradient projection (CGP) methods. [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] The inertial step is a strategy that involves using information from the previous two iterations to calculate the current iteration, and it is one of the most effective methods for accelerating the convergence of iterative methods. This idea was originally proposed by Polyak, 28 who was inspired by an implicit discretization of a second-order-in-time dissipative dynamical system, represented by…”

Section: Introductionmentioning

confidence: 99%

“…Due to their derivative‐free nature, the latter has attracted significant attention. Therefore, this work will only focus on the DFP methods, such as spectral gradient projection methods 7,8 and conjugate gradient projection (CGP) methods 9‐27 …”

Section: Introductionmentioning

confidence: 99%

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A family of inertial‐based derivative‐free projection methods with a correction step for constrained nonlinear equations and their applications

Liu,

Shao,

Yuan

et al. 2023

Numerical Linear Algebra App

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Numerous attempts have been made to develop efficient methods for solving the system of constrained nonlinear equations due to its widespread use in diverse engineering applications. In this article, we present a family of inertial‐based derivative‐free projection methods with a correction step for solving such system, in which the selection of the derivative‐free search direction is flexible. This family does not require the computation of corresponding Jacobian matrix or approximate matrix at every iteration and possess the following theoretical properties: (i) the inertial‐based corrected direction framework always automatically satisfies the sufficient descent and trust region properties without specific search directions, and is independent of any line search; (ii) the global convergence of the proposed family is proven under a weaker monotonicity condition on the mapping , without the typical monotonicity or pseudo‐monotonicity assumption; (iii) the results about convergence rate of the proposed family are established under slightly stronger assumptions. Furthermore, we propose two effective inertial‐based derivative‐free projection methods, each embedding a specific search direction into the proposed family. We present preliminary numerical experiments on certain test problems to demonstrate the effectiveness and superiority of the proposed methods in comparison with existing ones. Additionally, we utilize these methods for solving sparse signal restorations and image restorations in compressive sensing applications.

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Accelerated derivative‐free method for nonlinear monotone equations with an application

Ibrahim

Abubakar

Adamu

2021

Numerical Linear Algebra App

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In optimization theory, to speed up the convergence of iterative procedures, many mathematicians often use the inertial extrapolation method. In this article, based on the three‐term derivative‐free method for solving monotone nonlinear equations with convex constraints [Calcolo, 2016;53(2):133‐145], we design an inertial algorithm for finding the solutions of nonlinear equation with monotone and Lipschitz continuous operator. The convergence analysis is established under some mild conditions. Furthermore, numerical experiments are implemented to illustrate the behavior of the new algorithm. The numerical results have shown the effectiveness and fast convergence of the proposed inertial algorithm over the existing algorithm. Moreover, as an application, we extend this method to solve the LASSO problem to decode a sparse signal in compressive sensing. Performance comparisons illustrate the effectiveness and competitiveness of the method.

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A derivative-free three-term Hestenes–Stiefel type method for constrained nonlinear equations and image restoration

Cited by 16 publications

References 47 publications

Projection method with inertial step for nonlinear equations: Application to signal recovery

Projection method with inertial step for nonlinear equations: Application to signal recovery

A family of inertial‐based derivative‐free projection methods with a correction step for constrained nonlinear equations and their applications

Accelerated derivative‐free method for nonlinear monotone equations with an application

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