A family of three-term conjugate gradient projection methods with a restart procedure and their relaxed-inertial extensions for the 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

“…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…”

“…The research conducted in References 9 and 11 substantiates the relative advantages of incorporating an inertial step in methods designed to solve the nonlinear equations. Based on the three-term conjugate gradient (CG) method 33 and the accelerated gradient-descent method MSM, 34 Liu et al 10 developed a family of CGP methods with a restart strategy for solving constrained pseudo-monotone nonlinear equations, along with their relaxed-inertial version. The latter incorporates an acceleration technology similar to that in Reference 11, resulting in improved computational performance.…”

“…So, the search direction d * k always satisfies the sufficient descent property (10) and trust region property (11). Building on the discussions above, we employ a novel acceleration technique to enhance the numerical performance of the DFP method.…”