Spectral conjugate gradient projection methods for large-scale monotone equations without Lipschitz continuity

TL;DR

This paper presents two new derivative-free projection methods using spectral parameters for large-scale monotone equations, achieving convergence without Lipschitz continuity and demonstrating effectiveness through extensive numerical tests.

Contribution

The paper introduces two novel spectral conjugate gradient projection methods that do not require Lipschitz continuity for convergence in large-scale monotone equations.

Findings

Global convergence established for both methods.

Methods perform well on large-scale problems up to 120,000 dimensions.

Numerical experiments confirm practical effectiveness in signal recovery and logistic regression.

Abstract

We introduce two derivative-free projection methods for large-scale systems of nonlinear monotone equations subject to convex constraints. Both methods incorporate an adaptive spectral parameter into established conjugate gradient frameworks: the first generalizes the modified optimal Perry method via an eigenvalue-optimized scaling matrix, and the second generalizes the Hager--Zhang-type conjugate gradient projection method via a spectral Dai--Liao parameter. The resulting search directions satisfy a sufficient descent condition independent of the line search. For the first method, we establish global convergence under monotonicity alone, without requiring Lipschitz continuity of the mapping. For the second, global convergence holds under the standard monotonicity and Lipschitz continuity assumptions. Numerical experiments on 18 test problems across dimensions up to 120{,}000, together with applications to $\ell_1$-regularized signal recovery and regularized logistic regression, confirm the practical effectiveness of the proposed approach.