Derivative-free projection CG-based algorithm with restart strategy for solving convex-constrained nonlinear monotone equations and its application to logistic regression

This paper proposes a class of derivative-free projection algorithms with a restart technique for solving convex-constrained nonlinear equations involving monotone mappings. The proposed method integrates properties from classical conjugate gradient methods, such as the Polak–Ribière–Polyak, Liu–Storey, Fletcher–Reeves, and Conjugate-Descent methods. Second, it applies to nonsmooth equations, extending its utility to a broader range of problems. Third, the search direction of the new method is descent and bounded. Finally, numerical experiments are carried out on some test problems with the results provided to show the efficiency of the proposed method and to support theoretical analysis.