Systems Of Nonlinear Monotone Equations Research Articles

A derivative-free conjugate gradient method for large-scale nonlinear systems of monotone equations

A derivative-free conjugate gradient method for large-scale nonlinear systems of monotone equations

An extended projected residual algorithm for solving smooth convex optimization problems

A projected residual algorithm for solving smooth convex optimization problems is presented. The proposed method is an extension of a residual algorithm for solving systems of nonlinear monotone equations introduced by La Cruz (2017), which uses in a systematic way the residual as a search direction combined with the Barzilai–Borwein’s choice of the step size and a line search globalization strategy that does not impose the condition that the function value to decrease monotonically at every iteration. The global and R-sublinear convergence of the new method is established. With the aim of showing the advantages of the proposed global scheme an extensive set of numerical experiments including standard test problems and some specific applications are reported.

Using a New Projection Approach to Find the Optimal Solution for Nonlinear Systems of Monotone Equation

There are many algorithms are used to solve systems of nonlinear monotone equations with various advantages and disadvantages, including the line search algorithm, trust region algorithm, projection algorithm and others. In this paper we used a new projection algorithm to solve these systems. The projection methods are considered one of the effective free derivative methods to solve systems of nonlinear monotone equations. The framework of this method is that the current iterate is separated strictly from the solution set of the problem in each iteration by a suitable hyperplane which constructs by the new algorithm. Then, in order to determine the new approximation, the current iteration is projected on this hyperplane. The global convergence of the proposed algorithm is proven under standard assumptions. The numerical results showed that the suggested algorithm is very efficiency and promising.

Solving Systems of Nonlinear Monotone Equations by Using a New Projection Approach

The projection technique is one of the famous method and highly useful to solve the optimization problems and nonlinear systems of equations. In this work, a new projection approach for solving systems of nonlinear monotone equation is proposed combining with the conjugate gradient direction because of their low storage. The new algorithm can be used to solve the large-scale nonlinear systems of equations and satisfy the sufficient descent condition. The new algorithm generates appropriate direction then employs a good line search along this direction to reach a new point. If this point solves the problem then the algorithm stops, otherwise, it constructs a suitable hyperplane that strictly separate the current point from the solution set. The next iteration is obtained by projection the new point onto the separating hyperplane. We proved that the line search of the new projection algorithm is well defined. Furthermore, we established the global convergence under some mild conditions. The numerical experiment indicates that the new method is effective and very well.

A new projection technique for developing a Liu-Storey method to solve nonlinear systems of monotone equations

The projection technique is a very important method and efficient for solving unconstraint optimization and nonlinear equations. In this study, we developed a Liu-Story (LS) algorithm for solving monotone equations of nonlinear systems. The new algorithm satisfies the sufficient descent condition and it’s a suitable method of large scale equations for its limited memory. We established a global convergence of suggest method under the mild conditions. Numerical results proved that the new algorithm works well and promising.

A class of new derivative-free gradient type methods for large-scale nonlinear systems of monotone equations

In this paper, we present a class of new derivative-free gradient type methods for large-scale nonlinear systems of monotone equations. The methods combine the RMIL conjugate gradient method, the strategy of hyperplane projection and the derivative-free line search technique. Under some appropriate assumptions, the global convergence of the given methods is established. Numerical experiments indicate that the proposed methods are effective.

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 New Line Search Method to Solve the Nonlinear Systems of Monotone Equations

A New Line Search Method to Solve the Nonlinear Systems of Monotone Equations

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 spectral algorithm for large-scale systems of nonlinear monotone equations

A derivative-free iterative scheme that uses the residual vector as search direction for solving large-scale systems of nonlinear monotone equations is presented. It is closely related to two recently proposed spectral residual methods for nonlinear systems which use a nonmonotone line-search globalization strategy and a step-size based on the Barzilai-Borwein choice. The global convergence analysis is presented. In order to study the numerical behavior of the algorithm, it is included an extensive series of numerical experiments. Our computational experiments show that the new algorithm is computationally efficient.

A modified quasi‐Newton diagonal update algorithm for total variation denoising problems and nonlinear monotone equations with applications in compressive sensing

SummaryIn this paper, we present a new algorithm to accelerate the Chambolle gradient projection method for total variation image restoration. The new proposed method considers an approximation of the Hessian based on the secant equation. Combined with the quasi‐Cauchy equations and diagonal updating, we can obtain a positive definite diagonal matrix. In the proposed minimization method model, we use the positive definite diagonal matrix instead of the constant time stepsize in Chambolle's method. The global convergence of the proposed scheme is proved. Some numerical results illustrate the efficiency of this method. Moreover, we also extend the quasi‐Newton diagonal updating method to solve nonlinear systems of monotone equations. Performance comparisons show that the proposed method is efficient. A practical application of the monotone equations is shown and tested on sparse signal reconstruction in compressed sensing. Copyright © 2015 John Wiley & Sons, Ltd.

A new line search strategy for finding separating hyperplane in projection-based methods

Projection based methods are a family of efficient and applicable derivative free methods for solving systems of nonlinear monotone equations. These methods, at each iteration, use a backtracking line search to generate a hyperplane which strictly separates the current approximation from the solution set of the problem. Numerical experiments indicate that choosing an appropriate line search highly affects the efficiency of projection based methods. In this paper we introduce a new line search procedure for generating the separating hyperplane. The convergence properties of the new procedure is established in a simple and short way. Numerical results show that the new line search is very effective and increases the efficiency of projection based methods.

A double-projection-based algorithm for large-scale nonlinear systems of monotone equations

In this paper, we propose a derivative free algorithm for solving large-scale nonlinear systems of monotone equations which combines a new idea of projection methodology with a line search strategy, while an improvement in the projection step is exerted. At each iteration, the algorithm constructs two appropriate hyperplanes which strictly separate the current approximation from the solution set of the problem. Then the new approximation is determined by projecting the current point onto the intersection of two halfspaces that are constructed by these hyperplanes and contain the solution set of the problem. Under some mild conditions, the global convergence of the algorithm is established. Preliminary numerical results indicate that the proposed algorithm is promising.

An Adaptive Prediction-Correction Method for Solving Large-Scale Nonlinear Systems of Monotone Equations with Applications

Combining multivariate spectral gradient method with projection scheme, this paper presents an adaptive prediction-correction method for solving large-scale nonlinear systems of monotone equations. The proposed method possesses some favorable properties: (1) it is progressive step by step, that is, the distance between iterates and the solution set is decreasing monotonically; (2) global convergence result is independent of the merit function and its Lipschitz continuity; (3) it is a derivative-free method and could be applied for solving large-scale nonsmooth equations due to its lower storage requirement. Preliminary numerical results show that the proposed method is very effective. Some practical applications of the proposed method are demonstrated and tested on sparse signal reconstruction, compressed sensing, and image deconvolution problems.