A Modified Spectral PRP Conjugate Gradient Projection Method for Solving Large‐Scale Monotone Equations and Its Application in Compressed Sensing

Abstract

In this paper, we develop an algorithm to solve nonlinear system of monotone equations, which is a combination of a modified spectral PRP (Polak-Ribière-Polyak) conjugate gradient method and a projection method. The search direction in this algorithm is proved to be sufficiently descent for any line search rule. A line search strategy in the literature is modified such that a better step length is more easily obtained without the difficulty of choosing an appropriate weight in the original one. Global convergence of the algorithm is proved under mild assumptions. Numerical tests and preliminary application in recovering sparse signals indicate that the developed algorithm outperforms the state-of-the-art similar algorithms available in the literature, especially for solving large-scale problems and singular ones.It has been shown that the solution set of problem (1) is convex if it is nonempty [3]. In addition, throughout the paper, the space is equipped with the Euclidean norm ‖ ⋅ ‖ and the inner product ⟨ , ⟩ = , for , ∈ . Aiming at solution of problem (1), many efficient methods were developed recently. Only by incomplete enumeration, we here mention the trust region method [19], the Newton and the quasi-Newton methods [4][5][6]19], the Gauss-Newton methods [7,8], the Levenberg-Marquardt methods [20][21][22], the derivative-free methods and its modified versions [9][10][11][12][13][14][15][16][23][24][25][26][27], the derivative-free conjugate gradient projection method [14], the modified PRP (Polak-Ribière-Polyak) conjugate gradient method [11], the TPRP method [10], the PRP-type method [28], the projection method [23], the FR-type method [9], and the modified spectral conjugate gradient projection method [13]. Summarily, the spectral Hindawi

A dai-liao hybrid conjugate gradient method for unconstrained optimization

“…Esmaeili, et al (2018) suggested a new CG scheme to solve problems emerged from astronomical imaging, file restoration, image video coding and other applications. Similarly, Guo & Wan (2019) developed CG algorithm for sparse engineering signal problem. Numerical tests indicated that the algorithm is an alternative for recovering sparse signal problems and beats earlier methods.…”

A New Hybrid Conjugate Gradient Projection Method for Solving Nonlinear Monotone Equations

“…Recently, an adaptive line search approach that takes into consideration a disturbance component was presented by Liu et al [33] and Guo and Wan [34], i.e., the step size α k � max aρ i : i � 0, 1, . .…”

An Efficient Single-Parameter Scaling Memoryless Broyden-Fletcher-Goldfarb-Shanno Algorithm for Solving Large Scale Unconstrained Optimization Problems

“…Since a nonlinear system of equations is closely related with an optimization model, it is significant to extend the developed algorithm into solving large scale nonlinear system of equations from engineering fields. Actually, it has been shown [20], [22] that recovering sparse signals and restoring blurred images can be formulated as a system of equations, and efficient optimization algorithms can be modified to solve these practical engineering problems.…”

“…One of our goals in this paper is to find a better scaling parameter such that the corresponding algorithm is more efficient and robust. For many large-scale practical optimization models [22], [23], the (scaling) BFGS algorithms like (3) and ( 4) are often powerless because they are associated with solution of a large-scale system of linear equations B k d k = −g k , as well as computation and storage of matrices B k with large sizes. Therefore, another goal in this paper is to modify the scaling BFGS algorithms such that only gradient information in the optimization problem (1) is employed to compute the search directions and the step sizes without needs of computing and storing matrices [15].…”