A Derivative-Free Method with the Symmetric Rank-One Update for Constrained Nonlinear Systems of Monotone Equations

This paper presents a new derivative-free method with a symmetric rank-one (SR1) update for handling constrained nonlinear systems of monotone equations. Distinctively, the approach employs a revised SR1 update formula that leverages information from the latest three iterates and their corresponding function values—a key improvement over existing SR1 variants relying only on two-point information. Derived from this revised formula, the approach integrates projection techniques for performance enhancement. One of its key advantages lies in the search direction: it exhibits sufficient descent and trust-region properties while eliminating the need for line search techniques. Theoretical analysis shows that the algorithm converges globally under specified conditions. Furthermore, numerical experiments are conducted to compare the proposed algorithm with two other existing approaches; results demonstrate that it exhibits superior reliability and robustness in terms of the number of iterations, function evaluations, and CPU runtime.