Two Generalized Derivative-free Methods to Solve Large Scale Nonlinear Equations with Convex Constraints
Kabenge Hamiss, Mohammed M. Alshahrani, Mujahid N. Syed
TL;DR
The paper tackles solving large-scale nonlinear equations $G(x)=0$ under convex constraints $x\in\Gamma$ with derivative-free methods. It introduces two generalized solvers, GMOPCGM and GCGPM, that extend the Modified Optimal Perry Conjugate Gradient Method and the Conjugate Gradient Projection Method by adaptive parameterization and spectral-like updates, while ensuring a sufficient descent condition. Theoretical results establish global convergence under standard Lipschitz assumptions (and variants without Lipschitz continuity via line search), with detailed algorithmic formulations and convergence proofs. Numerical experiments on 19 problems, including signal restoration via compressed sensing, demonstrate that the generalized methods outperform or closely match existing approaches in iterations, function evaluations, and CPU time, with GCGPM showing robust performance across metrics.
Abstract
In this work, we propose two derivative-free methods to address the problem of large-scale nonlinear equations with convex constraints. These algorithms satisfy the sufficient descent condition. The search directions can be considered generalizations of the Modified Optimal Perry conjugate gradient method and the conjugate gradient projection method or the Spectral Modified Optimal Perry conjugate gradient method and the Spectral Conjugate Gradient Projection method. The global convergence of the former does not depend on the Lipschitz continuity of G. In contrast, the latter's global convergence depends on the Lipschitz continuity of G. The numerical results show the efficiency of the algorithms.