Spectral conjugate gradient projection methods for large-scale monotone equations without Lipschitz continuity
This work provides computationally efficient solvers for large-scale monotone equations, particularly valuable when Lipschitz continuity is unavailable, though the novelty is incremental.
Two derivative-free projection methods for large-scale monotone equations with convex constraints are introduced, incorporating adaptive spectral parameters into conjugate gradient frameworks. The methods achieve global convergence without Lipschitz continuity (first method) and demonstrate effectiveness on 18 test problems up to dimension 120,000, with applications to signal recovery and logistic regression.
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.