An Efficient Conjugate Gradient Method for Convex Constrained Monotone Nonlinear Equations with Applications

Abstract

This research paper proposes a derivative-free method for solving systems of nonlinearequations with closed and convex constraints, where the functions under consideration are continuousand monotone. Given an initial iterate, the process first generates a specific direction and then employsa line search strategy along the direction to calculate a new iterate. If the new iterate solves theproblem, the process will stop. Otherwise, the projection of the new iterate onto the closed convex set(constraint set) determines the next iterate. In addition, the direction satisfies the sufficient descentcondition and the global convergence of the method is established under suitable assumptions.Finally, some numerical experiments were presented to show the performance of the proposedmethod in solving nonlinear equations and its application in image recovery problems.

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

“…Furthermore, F was proved to be continuous and monotone in [46]. Therefore problem (25) can be translated into problem (1) and thus MFRM method can be applied to solve it.…”

“…It is important to mention that nonlinear monotone equations arise in many practical applications. These and other reasons motivate researchers to develop a large number of class of Iterative methods for solving such systems, for example, see [1][2][3][4][5][6][7] among others. In addition, convex constrained equations have application in many scientific fields, some of which are the economic equilibrium problems [8], the chemical equilibrium systems [9], etc.…”

“…In addition, convex constrained equations have application in many scientific fields, some of which are the economic equilibrium problems [8], the chemical equilibrium systems [9], etc. Several algorithms were developed to solve (1), among them, are the trust-region [10] and the Levenberg-Marquardt method [11]. Moreover, the requirement to compute and store the matrix in every iteration makes them ineffective for large-scale nonlinear equations.…”

“…For examples, Xiao and Zhu [20] presented a CG method, which combines the well-known CG-DESCENT method in [17] and the projection method by Solodov and Svaiter [21]. Liu et al [22] proposed two CG methods with projection strategy for solving (1). In [23], a modification of the method in [20] was presented by Liu and Li.…”

“…In addition, a hybrid CG projection method for convex constrained equations was developed in [25]. Ou and Li [26] proposed a combination of a scaled CG method and the projection strategy to solve (1). Furthermore, Ding et al [27] extended the Dai and Kou (DK) CG method to solve (1) by also combining it with the projection method.…”