A Modified Scaled Spectral-Conjugate Gradient-Based Algorithm for Solving Monotone Operator Equations

Abstract

This paper proposes a modified scaled spectral-conjugate-based algorithm for finding solutions to monotone operator equations. The algorithm is a modification of the work of Li and Zheng in the sense that the uniformly monotone assumption on the operator is relaxed to just monotone. Furthermore, unlike the work of Li and Zheng, the search directions of the proposed algorithm are shown to be descent and bounded independent of the monotonicity assumption. Moreover, the global convergence is established under some appropriate assumptions. Finally, numerical examples on some test problems are provided to show the efficiency of the proposed algorithm compared to that of Li and Zheng.

Projection method with inertial step for nonlinear equations: Application to signal recovery

“…Considering the simplicity and low storage requirement of the conjugate gradient method [20,21], several researchers combined the projection technique of Solodov and Svaiter [56] with the conjugate gradient methods to solve large-scale nonlinear equations, see [38,13,42,40,43,1,7,45,41,12,47,10,9,46,39,52,2,3,8,53,11,37,44,4,6,36] and references therein. Based on the projection method, Gao and He [35] introduced an efficient three-term derivative-free method for solving nonlinear monotone equations with convex constraints (1) by choosing a part of the Liu-Storey (LS) conjugate parameter as a new conjugate parameter.…”

"A descent three-term derivative-free method for signal reconstruction in compressive sensing"

“…We compare the performance of DSTT with the SGCS [37], CGD [38] and PCG [34] algorithms designed for similar purpose. In the experiment, we consider a signal of size m = 2 12 , n = 2 10 and the original signal contains 2 6 randomly nonzero elements. The random T is the Gaussian matrix which is generated by command randn(m,n) in Matlab.…”

“…Exploiting the simplicity and low storage requirement of the conjugate gradient method [1,2], in recent times, several authors have extended many conjugate gradient algorithms designed to solve unconstrained optimization problems to solve large-scale nonlinear equations (1.6) (see [3][4][5][6][7][8][9][10][11][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]36]). For instance, using the projection scheme of Solodov and Svaiter [35], Xiao and Zhu [38] extended the Hager and Zhang conjugate descent (CG DESCENT) method to solve (1.6).…”

Accelerated derivative‐free method for nonlinear monotone equations with an application

“…More precisely, the CG method has been paid attention to as an effective numerical method for solving large‐scale unconstrained optimization problems because of its simplicity and low storage 7,8 . Thus, using the projection technique in Solodov and Svaiter, 9 several researchers extended these methods resulting in derivative‐free methods (see References 10‐33 and references therein). Recently, based on Hestenes–Stiefel (HS) CG method 34 for unconstrained optimization, Wang et al 35 proposed a self‐adaptive three‐term derivative‐free method for solving monotone nonlinear equations with convex constraints.…”