Two Hybrid Spectral Methods With Inertial Effect for Solving System of Nonlinear Monotone Equations With Application in Robotics

Abstract

In this paper, 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 1 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.

On efficient matrix-free method via quasi-Newton approach for solving system of nonlinear equations

“…Hence for ∈ (0, 1) we have (22). Since the search direction satised the decent condition in (22), it means that the inequality F k+1 ≤ F k holds. Proof.…”

Alternative structured spectral gradient algorithms for solving nonlinear least-squares problems

“…Furthermore, the implementation of Algorithm 1 using , that is, the geometric mean of and , proved to be more efficient as it successfully solved all the test problems considered in our numerical experiments without any failure. Our future study will include using a two-step approach of [29] to solve non-linear least-squares problems and applying these algorithms to motion control problems [30] , [31] , [32] .…”

Survey of Methods for Solving Systems of Nonlinear Equations, Part I: Root-finding Approaches

“…Spectral methods are another class of derivative-free methods which have been used to solve SNEs, and they are particularly well suited for solving large SNEs because they have a relatively low storage requirement [59]. Spectral methods often utilize spectral parameters and coefficients to help determine the search direction d k in the update step x k+1 = x k + α k d k .…”

“…Many of the root finding methods described in Section 3 are guaranteed to converge to a solution when applied to SNEs that satisfy both the monotonicity and Lipchitz continuity conditions (the hybrid spectral methods introduced in [59] for example).…”