A unified derivative-free projection method model for large-scale nonlinear equations with convex constraints

Abstract

<p style='text-indent:20px;'>Motivated by recent derivative-free projection methods proposed in the literature for solving nonlinear constrained equations, in this paper we propose a unified derivative-free projection method model for large-scale nonlinear equations with convex constraints. Under mild conditions, the global convergence and convergence rate of the proposed method are established. In order to verify the feasibility and effectiveness of the model, a practical algorithm is devised and the corresponding numerical experiments are reported, which show that the proposed practical method is efficient and can be applied to solve large-scale nonsmooth equations. Moreover, the proposed practical algorithm is also extended to solve the obstacle problem.</p>

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

“…Remark The Assumption (A3) was originally introduced by Solodov and Svaiter 37 to establish the global convergence of projection method for variational inequality problem. Later on, it was used by References 23,38,39 to prove the global convergence of the projection methods for constrained nonlinear equations. It should be noted that Assumption (A3) is satisfied if the mapping $$$ E $$$ is monotone, that is, $$$$ {\left[E(x)-E(y)\right]}^{\top}\left(x-y\right)\ge 0,\kern0.3em \forall \kern0.3em x,\kern0.3em y\in {\mathbb{R}}^n,\kern1.00em $$$$ or pseudo‐monotone, that is, $$$$ E{(y)}^{\top}\left(x-y\right)\ge 0\Rightarrow E{(x)}^{\top}\left(x-y\right)\ge 0,\kern0.3em \forall \kern0.3em x,\kern0.3em y\in {\mathbb{R}}^n, $$$$ but not vice versa (see Reference 40 for more details).…”