Nonlinear Monotone Equations Research Articles

A hybrid conjugate gradient algorithm for constrained monotone equations with application in compressive sensing

Combining the projection method of Solodov and Svaiter with the Liu-Storey and Fletcher Reeves conjugate gradient algorithm of Djordjević for unconstrained minimization problems, a hybrid conjugate gradient algorithm is proposed and extended to solve convex constrained nonlinear monotone equations. Under some suitable conditions, the global convergence result of the proposed method is established. Furthermore, the proposed method is applied to solve the ℓ1-norm regularized problems to restore sparse signal and image in compressive sensing. Numerical comparisons of the proposed algorithm versus some other conjugate gradient algorithms on a set of benchmark test problems, sparse signal reconstruction and image restoration in compressive sensing show that the proposed scheme is computationally more efficient and robust than the compared schemes.

A Modified Hestenes-Stiefel-Type Derivative-Free Method for Large-Scale Nonlinear Monotone Equations

The goal of this paper is to extend the modified Hestenes-Stiefel method to solve large-scale nonlinear monotone equations. The method is presented by combining the hyperplane projection method (Solodov, M.V.; Svaiter, B.F. A globally convergent inexact Newton method for systems of monotone equations, in: M. Fukushima, L. Qi (Eds.)Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Academic Publishers. 1998, 355-369) and the modified Hestenes-Stiefel method in Dai and Wen (Dai, Z.; Wen, F. Global convergence of a modified Hestenes-Stiefel nonlinear conjugate gradient method with Armijo line search. Numer Algor. 2012, 59, 79-93). In addition, we propose a new line search for the derivative-free method. Global convergence of the proposed method is established if the system of nonlinear equations are Lipschitz continuous and monotone. Preliminary numerical results are given to test the effectiveness of the proposed method.

Self adaptive spectral conjugate gradient method for solving nonlinear monotone equations

In this paper, we propose a self adaptive spectral conjugate gradient-based projection method for systems of nonlinear monotone equations. Based on its modest memory requirement and its efficiency, the method is suitable for solving large-scale equations. We show that the method satisfies the descent condition F_{k}^{T}d_{k}leq -c|F_{k}|^{2}, c>0, and also prove its global convergence. The method is compared to other existing methods on a set of benchmark test problems and results show that the method is very efficient and therefore promising.

A Modified Conjugate Descent Projection Method for Monotone Nonlinear Equations and Image Restoration

In this article, we propose a modified conjugate descent (CD) projection algorithm for solving system of nonlinear monotone equations with convex constraints. The search direction in this algorithm use a convex combination of the steepest descent algorithm and the well-known CD method. The algorithm proves to be quite efficient for solving large scale monotone nonlinear equations, as it has low storage requirement and does not need the computation of Jacobian matrix. We prove the convergence of the algorithm using some conditions, and perform numerical experiments on some test problems. In order to show the efficiency of our proposed algorithm, the numerical performance is compared with some existing algorithms. Finally, by reformulating l 2 regularized problem into monotone equation, we successfully apply the algorithm to restore some blurred images. The numerical results obtained prove that the algorithm can be used as an efficient and qualitative solver for image restoration problems.

A Modified Spectral Gradient Projection Method for Solving Non-linear Monotone Equations with Convex Constraints and Its Application

In this paper, we propose a derivative free algorithm for solving non-linear monotone equations with convex constraints. The proposed algorithm combines the method of spectral gradient and the projection method. We also modify the backtracking line search technique. The global convergence of the proposed method is guaranteed, under the mild conditions. Further, the numerical experiments show that the large-scale non-linear equations with convex constraints can be effectively solved with our method. The $L_{1}$ -norm regularized problems in signal reconstruction are studied by using our method.

Inertial-Based Derivative-Free Method for System of Monotone Nonlinear Equations and Application

Iterative methods for solving nonlinear problems are of great importance due to their appearance in various areas of applications. In this paper, based on the inertial effect, we propose two projection derivative-free iterative methods for solving system of nonlinear equations. For the purpose of improving the numerical performance, the two methods incorporated the inertial step into the modified Barzilai and Borwein (BB) spectral parameters to generate the sequence of their respective search directions. The two spectral parameters are shown to be well-defined. For each method, the sequence of the search direction is bounded and satisfies the sufficient descent property. We establish the convergence analysis of the two methods based on the assumption that the underlying mapping is Lipschitzian and monotone. We demonstrate the efficiencies of the two methods on some collection of monotone system of nonlinear equations test problems. Finally, we apply the two methods to solve motion control problem involving a two planar robot.

A NEW THREE-TERM CONJUGATE GRADIENT-BASED PROJECTION METHOD FOR SOLVING LARGE-SCALE NONLINEAR MONOTONE EQUATIONS

A new three-term conjugate gradient-based projection method is presented in this paper for solving large-scale nonlinear monotone equations. This method is derivative-free and it is suitable for solving large-scale nonlinear monotone equations due to its lower storage requirements. The method satisfies the sufficient descent condition FTkdk ≤ −τ‖Fk‖2, where τ > 0 is a constant, and its global convergence is also established. Numerical results show that the method is efficient and promising.

A diagonal PRP-type projection method for convex constrained nonlinear monotone equations

<p style='text-indent:20px;'>Iterative methods for nonlinear monotone equations do not require the differentiability assumption on the residual function. This special property of the methods makes them suitable for solving large-scale nonsmooth monotone equations. In this work, we present a diagonal Polak-Ribi<inline-formula><tex-math id="M1">\begin{document}$ \grave{e} $\end{document}</tex-math></inline-formula>re-Polyak (PRP) conjugate gradient-type method for solving large-scale nonlinear monotone equations with convex constraints. The search direction is a combine form of a multivariate (<i>diagonal</i>) spectral method and a modified PRP conjugate gradient method. Proper safeguards are devised to ensure positive definiteness of the diagonal matrix associated with the search direction. Based on Lipschitz continuity and monotonicity assumptions the method is shown to be globally convergent. Numerical results are presented by means of comparative experiments with recently proposed multivariate spectral Dai-Yuan-type (J. Ind. Manag. Optim. 13 (2017) 283-295) and Wei-Yao-Liu-type (Int. J. Comput. Math. 92 (2015) 2261-2272) conjugate gradient methods.

Global convergence via descent modified three-term conjugate gradient projection algorithm with applications to signal recovery

In this article, we propose a three-term conjugate gradient projection algorithm for solving constrained monotone nonlinear equations. The global convergence of the algorithm was established under suitable assumptions. Numerical examples presented indicate that the algorithm has a very good performance in solving monotone nonlinear equations. Finally, the algorithm is applied to solve signal recovery problems.

An efficient hybrid conjugate gradient-based projection method for convex constrained nonlinear monotone equations

Conjugate gradient projection methods are widely used for solving large-scale nonlinear systems of monotone equations because of their low memory requirements and research continues to be done to improve these methods. In this paper, we introduce a new hybrid conjugate gradient projection method for solving large-scale nonlinear monotone equations. This proposed method is shown to satisfy the descent property and its global convergence is also established. Numerical results show that the new method is very much promising when compared with other conjugate gradient projection methods.

A Modified Fletcher–Reeves Conjugate Gradient Method for Monotone Nonlinear Equations with Some Applications

One of the fastest growing and efficient methods for solving the unconstrained minimization problem is the conjugate gradient method (CG). Recently, considerable efforts have been made to extend the CG method for solving monotone nonlinear equations. In this research article, we present a modification of the Fletcher–Reeves (FR) conjugate gradient projection method for constrained monotone nonlinear equations. The method possesses sufficient descent property and its global convergence was proved using some appropriate assumptions. Two sets of numerical experiments were carried out to show the good performance of the proposed method compared with some existing ones. The first experiment was for solving monotone constrained nonlinear equations using some benchmark test problem while the second experiment was applying the method in signal and image recovery problems arising from compressive sensing.

A Modified Self-Adaptive Conjugate Gradient Method for Solving Convex Constrained Monotone Nonlinear Equations for Signal Recovery Problems

In this article, we propose a modified self-adaptive conjugate gradient algorithm for handling nonlinear monotone equations with the constraints being convex. Under some nice conditions, the global convergence of the method was established. Numerical examples reported show that the method is promising and efficient for solving monotone nonlinear equations. In addition, we applied the proposed algorithm to solve sparse signal reconstruction problems.

Spectral modified Polak–Ribiére–Polyak projection conjugate gradient method for solving monotone systems of nonlinear equations

In this paper, we present a modification of Polak–Ribie´re–Polyak (PRP) conjugate gradient method for solving system of monotone nonlinear equations which is a combination of spectral conjugate gradient method and the hyperplane projection technique. The method is based on two methods for unconstrained optimization proposed by Wan et al. (2011) and Sun (2015). We obtained a new search direction by the use of a different formula for the conjugate gradient parameter. The search direction satisfies the sufficient descent condition and the global convergence of the method is established under some assumptions. Preliminary numerical comparison with some existing methods shows the efficiency of the proposed method.

A family of Hager–Zhang conjugate gradient methods for system of monotone nonlinear equations

This paper presents two modified Hager–Zhang (HZ) Conjugate Gradient methods for solving large-scale system of monotone nonlinear equations. The methods were developed by combining modified forms of the one-parameter method by Hager and Zhang (2006) and the hyperplane projection technique. Global convergence and numerical results of the methods are established. Preliminary numerical results show that the proposed methods are promising and more efficient compared to the methods presented by Mushtak and Keyvan (2018) and Sun et al. (2017).

A modified conjugate gradient method for monotone nonlinear equations with convex constraints

In this paper, a modified Hestenes-Stiefel (HS) spectral conjugate gradient (CG) method for monotone nonlinear equations with convex constraints is proposed based on projection technique. The method can be viewed as an extension of a modified HS-CG method for unconstrained optimization proposed by Amini et al. (Optimization Methods and Software, pp: 1-13, 2018). A new search direction is obtained by incorporating the idea of spectral gradient parameter and some modification of the conjugate gradient parameter. The proposed method is derivative-free and requires low memory which makes it suitable for large scale monotone nonlinear equations. Global convergence of the method is established under suitable assumptions. Preliminary numerical comparisons with some existing methods are given to show the efficiency of our proposed method. Furthermore, the proposed method is successfully applied to solve sparse signal reconstruction in compressive sensing.

An adaptive family of projection methods for constrained monotone nonlinear equations with applications

An adaptive class of conjugate gradient methods is proposed in this paper, which all possess descent property under strong-wolfe line search. The adaptive parameter in the search direction is determined by minimizing the distance between relative matrix and self-scaling memoryless BFGS update by Oren in the Frobenius norm. Two formulas of the adaptive parameter are further obtained, which are presented as those given by Oren and Luenberger (1973/74) and respectively Oren and Spedicato (1976). By projection technology, two adaptive projection algorithms are developed for solving monotone nonlinear equations with convex constraints. Some numerical comparisons show that these two algorithms are efficient. Last, the proposed algorithms are used to recover a sparse signal from incomplete and contaminated sampling measurements; the results are promising.

An efficient three-term conjugate gradient method for nonlinear monotone equations with convex constraints

In this paper, based on the hyperplane projection technique, we propose a three-term conjugate gradient method for solving nonlinear monotone equations with convex constraints. Due to the derivative-free feature and lower storage requirement, the proposed method can be applied to the solution of large-scale non-smooth nonlinear monotone equations. Under some mild assumptions, the global convergence is proved when the line search fulfils the backtracking line search condition. Moreover, we prove that the proposed method is R-linearly convergent. Numerical results show that our method is competitive and efficient for solving large-scale nonlinear monotone equations with convex constraints.

Partially symmetrical derivative-free Liu–Storey projection method for convex constrained equations

ABSTRACTApplying Powell symmetrical technique to the Liu–Storey conjugate gradient method, a partially symmetrical Liu–Storey conjugate gradient method is proposed and extended to solve nonlinear monotone equations with convex constraints, which satisfies the sufficient descent condition without any line search. By using some line searches, the global convergence is proved merely by assuming that the equations are Lipschitz continuous. Moreover, we prove the R-linear convergence rate of the proposed method with an additional assumption. Finally, compared with one existing method, the performance of the proposed method is showed by some numerical experiments on the given test problems.

A derivative-free iterative method for nonlinear monotone equations with convex constraints

In this paper, based on the projection strategy, we propose a derivative-free iterative method for large-scale nonlinear monotone equations with convex constraints, which can generate a sufficient descent direction at each iteration. Due to its lower storage and derivative-free information, the proposed method can be used to solve large-scale non-smooth problems. The global convergence of the proposed method is proved under the Lipschitz continuity assumption. Moreover, if the local error bound condition holds, the proposed method is shown to be linearly convergent. Preliminary numerical comparison shows that the proposed method is efficient and promising.

A derivative-free three-term projection algorithm involving spectral quotient for solving nonlinear monotone equations

ABSTRACTIn this paper, we develop a three-term conjugate gradient method involving spectral quotient, which always satisfies the famous Dai-Liao conjugacy condition and quasi-Newton secant equation, independently of any line search. This new three-term conjugate gradient method can be regarded as a variant of the memoryless Broyden-Fletcher-Goldfarb-Shanno quasi-Newton method with regard to spectral quotient. By combining this method with the projection technique proposed by Solodov and Svaiter in 1998, we establish a derivative-free three-term projection algorithm for dealing with large-scale nonlinear monotone system of equations. We prove the global convergence of the algorithm and obtain the R-linear convergence rate under some mild conditions. Numerical results show that our projection algorithm is effective and robust, and is more competitive with the TTDFP algorithm proposed Liu and Li [A three-term derivative-free projection method for nonlinear monotone system of equations. Calcolo. 2016;53:427–450].