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

Abstract

In this work, by considering the hyperplane projection and hybrid techniques, three scaled three-term conjugate gradient methods are extended to solve the system of constrained monotone nonlinear equations, and the developed methods have the advantages of low storage and only using function values. The new methods satisfy the sufficient descent condition independent of any line search criterion. It has been proved that three new methods converge globally under some mild conditions. The numerical experiments for constrained monotone nonlinear equations and image de-blurring problems illustrate that the proposed methods are numerically effective and efficient.

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

“…This has motivated the use of an adaptive line search (16), proposed in Reference 20, to compute the step-length 𝜂 k . It is worth noting that ( 16) is a generalization of ( 20) and (21). As long as the inertial-based corrected direction d † k satisfies the descent property, the adaptive line search ( 16) is well-defined, which we prove by contradiction.…”

“…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 …”