A family of inertial derivative-free projection methods for constrained nonlinear pseudo-monotone equations with applications

A family of inertial‐based derivative‐free projection methods with a correction step for constrained nonlinear equations and their applications

“…The step $$$ {\alpha}_k\in \left[0,1\right) $$$ derived from this study theoretically covers all the cases mentioned in References 9‐16 above.…”

“…The codes are written in Matlab R2021b 64‐bit. For better comparison, we implement two noninertial methods, that is, HCGPM 24 (Algorithm 3.1) and PDY 25 (Algorithm 2.1) , as well as two inertial methods, that is, IHCGPM 12 (Algorithm 1) and IRTTCGPM 13 (Algorithm 2.2) . In the two noninertial methods, the inertial technique is not utilized, and they are conventional DFP methods.…”

“…The iterative process starts at the measurement signal $$$ {x}_0={A}^{\top }b $$$, and terminates when the relative change between successive iterative falls below $$$ 1{0}^{-5} $$$, that is, $$$$ \frac{\mid \triangle \left({x}_k\right)-\triangle \left({x}_{k-1}\right)\mid }{\mid \triangle \left({x}_{k-1}\right)\mid }<1{0}^{-5}. $$$$ Now, the csIAHDFPM and csILBFGS are applied to solve the sparse signal restoration problems, and we compare them with the PDY 25 (Algorithm 2.1) , IHCGPM 12 (Algorithm 1) and IRTTCGPM 13 (Algorithm 2.2) , whose parameters are the same as their original papers, respectively. The codes are written in Matlab R2021b 64‐bit.…”

“…Therefore, this work will only focus on the DFP methods, such as spectral gradient projection methods 7,8 and conjugate gradient projection (CGP) methods. [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] The inertial step is a strategy that involves using information from the previous two iterations to calculate the current iteration, and it is one of the most effective methods for accelerating the convergence of iterative methods. This idea was originally proposed by Polyak, 28 who was inspired by an implicit discretization of a second-order-in-time dissipative dynamical system, represented by…”

“…Due to their derivative‐free nature, the latter has attracted significant attention. Therefore, this work will only focus on the DFP methods, such as spectral gradient projection methods 7,8 and conjugate gradient projection (CGP) methods 9‐27 …”