Abstract
In this paper, we propose a two-step iterative algorithm based on projection technique for solving system of monotone nonlinear equations with convex constraints. The proposed two-step algorithm uses two search directions which are defined using the well-known Barzilai and Borwein (BB) spectral parameters.The BB spectral parameters can be viewed as the approximations of Jacobians with scalar multiple of identity matrices. If the Jacobians are close to symmetric matrices with clustered eigenvalues then the BB parameters are expected to behave nicely. We present a new line search technique for generating the separating hyperplane projection step of Solodov and Svaiter (1998) that generalizes the one used in most of the existing literature. We establish the convergence result of the algorithm under some suitable assumptions. Preliminary numerical experiments demonstrate the efficiency and computational advantage of the algorithm over some existing algorithms designed for solving similar problems. Finally, we apply the proposed algorithm to solve image deblurring problem.
Highlights
Many problems arising from various applications such as optimization, differential equations, variational inequalities problems and so on, can be converted into nonlinear system of equations. the study of iterative algorithms for solving nonlinear equations is of paramount importance especially when analytical method is not feasible or difficult to implement.Let F : Rn → Rn be a monotone mapping and Λ be a subset of Rn
The two existing methods are: (i) Spectral gradient projection method for monotone nonlinear equations with convex constraints proposed by Yu et al [17]
This paper presents an efficient derivative-free iterative algorithm called Two-Step Spectral Gradient Projection Method (TSSP) for nonlinear monotone equations
Summary
Many problems arising from various applications such as optimization, differential equations, variational inequalities problems and so on, can be converted into nonlinear system of equations. The BB method with the stepsize τkBB1 has been extended to solve unconstrained nonlinear equations by La Cruz and Raydan [13] Their algorithm is built on the strategy of nonmonotone line search technique which guarantees the global convergence of the method. In Reference [17], Yu et al extended the method given by Zhang and Zhou [16] to solve monotone system of nonlinear equations with convex constraints Their method is globally convergent under some conditions and preliminary numerical results show that the method works well and is more suitable compared to the projection method in Reference [18].
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