A diagonal PRP-type projection method for convex constrained nonlinear monotone equations

TL;DR

This paper introduces a diagonal PRP-type conjugate gradient method tailored for large-scale convex constrained nonlinear monotone equations, ensuring global convergence without requiring residual function differentiability.

Contribution

The paper proposes a novel diagonal PRP conjugate gradient method combining spectral and PRP techniques, with safeguards for positive definiteness, applicable to nonsmooth large-scale problems.

Findings

Method is globally convergent under Lipschitz and monotonicity assumptions.

Numerical experiments show competitive performance with existing spectral methods.

The approach effectively handles nonsmooth, large-scale convex constrained problems.

Abstract

Iterative methods for nonlinear monotone equations do not require the differentiability assumption on the residual function. This special property of the methods makes them suitable for solving large-scale nonsmooth monotone equations. In this work, we present a diagonal Polak-Ribeire-Polyk (PRP) conjugate gradient-type method for solving large-scale nonlinear monotone equations with convex constraints. The search direction is a combine form of a multivariate (diagonal) spectral method and a modified PRP conjugate gradient method. Proper safeguards are devised to ensure positive definiteness of the diagonal matrix associated with the search direction. Based on Lipschitz continuity and monotonicity assumptions the method is shown to be globally convergent. Numerical results are presented by means of comparative experiments with a recently proposed multivariate spectral conjugate gradient-type method.