Derivative-Free Spectral Projection Methods for Large-Scale Monotone Equations

We introduce two derivative-free spectral projection methods for large-scale monotone equations with convex constraints. The first, SOPP (Spectral Optimal-Perry Projection), selects its Perry parameter by minimizing the condition number of a symmetrized Perry matrix over its positive definite range, in place of the eigenvalue-gap criterion used in earlier work. A clipping step gives its search direction a trust-region property by construction. The second, SDLP (Spectral Dai–Liao Projection), replaces a fixed Dai–Liao factor with an adaptive spectral parameter. Both directions satisfy sufficient descent independently of the line search. Under standard assumptions each method either terminates finitely at a solution or generates a whole sequence converging to one. The SOPP result requires no Lipschitz continuity, whereas the SDLP analysis does. Numerical experiments on benchmark problems and two applications indicate that both methods are computationally viable and stable under reasonable parameter choices.