Descent Derivative-Free Method Involving Symmetric Rank-One Update for Solving Convex Constrained Nonlinear Monotone Equations and Application to Image Recovery

Abstract

Many practical applications in applied sciences such as imaging, signal processing, and motion control can be reformulated into a system of nonlinear equations with or without constraints. In this paper, a new descent projection iterative algorithm for solving a nonlinear system of equations with convex constraints is proposed. The new approach is based on a modified symmetric rank-one updating formula. The search direction of the proposed algorithm mimics the behavior of a spectral conjugate gradient algorithm where the spectral parameter is determined so that the direction is sufficiently descent. Based on the assumption that the underlying function satisfies monotonicity and Lipschitz continuity, the convergence result of the proposed algorithm is discussed. Subsequently, the efficiency of the new method is revealed. As an application, the proposed algorithm is successfully implemented on image deblurring problem.

The global convergence of spectral RMIL conjugate gradient method for unconstrained optimization with applications to robotic model and image recovery

“…Consider the sequence {x k } generated by spectral CG method Eqs (2) and (7). Pre-multiplying Eq (7) by g T k and using Eq (8) together with Cauchy-Schwartz inequality gives…”

“…Step 3: Compute d k by Eq (7), where β k and θ k are determined by Eqs ( 8) and ( 13), respectively.…”

“…R is smooth, meaning its gradient, g(x) = rf(x) is obtainable. Problems in the form of Eq (1) attracted much attention in optimization community because of their wide variety of applications in different fields [1], including portfolio choice [2,3], m-tensor system [4], image restoration [5][6][7][8], signal recovery [9][10][11] and robotic motion [12][13][14] among others. The iterative methods that make use of gradient of f are usually preferable to solve these problems.…”

A Modified Three-Term Conjugate Descent Derivative-Free Method for Constrained Nonlinear Monotone Equations and Signal Reconstruction Problems

“…However, they require computing and storing of the inverse Jacobian matrix at every iteration, which is not always easy to compute or not available due to singularity. As such, many researchers have suggested derivative-free approaches that do not require computation of the Jacobian matrix; among others are [16][17][18][19][20][21][22][23][24][25]. A modified three-term conjugate descent (CD) derivative-free method is of interest in this research.…”