A modified inertial three-term conjugate gradient projection method for constrained nonlinear equations with applications in compressed sensing

Three adaptive hybrid derivative‐free projection methods for constrained monotone nonlinear equations and their applications

“…Finally, similar to References 36‐38, we think that it could be interesting to construct some more efficient derivative‐free projection methods based on inertial extrapolation technique.…”

“…In fact, the CGM has been attached much importance as an effective numerical method for solving large‐scale unconstrained optimization problems because of its simplicity and low storage 8‐29 . Thus, using the projection technique, 30 several researchers extended these methods resulting in derivative‐free methods (see References 31‐48 and references therein). For example, based on Hestenes–Stiefel (HS) method 15 for unconstrained optimization, Wang et al 43 proposed a self‐adaptive three‐term conjugate gradient projection method (CGPM) for solving the SCMNE.…”

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…”

“…where || ⋅ || stands for the Euclidean norm on R n . Using the three-term CGP method developed by Reference 18, Ma et al 11 introduced a modified inertial three-term CGP method that integrates the inertial extrapolation step into the process of selecting the search direction, and the resulting direction satisfies the sufficient descent property without relying on any line search rule. Notably, the convergence analysis is conditioned on the following requirement,…”

“…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.…”