Broyden quasi-Newton secant-type method for solving constrained mixed generalized equations
P. C. da Silva Junior, O. P. Ferreira, G. N. Silva
TL;DR
The paper develops a novel variant of the Broyden quasi-Newton secant-type method for solving constrained mixed generalized equations of the form $f(x) + g(x) + F(x) \ni 0$ with $x \in C$, where $f$ is differentiable, $g$ may be non-differentiable, and $F$ is set-valued. It combines a secant-type Jacobian update with a Conditional Gradient projection to enforce feasibility and handle constraints, using divided differences to approximate derivatives. Under Lipschitz continuity of $f'$, bounded second-order divided differences of $g$, and metric regularity of the generalized equation, the method is proven to generate a well-defined sequence that converges locally at a Q-linear rate; strong metric regularity yields uniqueness of the limit. The results extend Newton-type methods for generalized equations by integrating constraint handling and derivative-free updates, with potential impact on constrained variational inequalities and complementarity problems.
Abstract
This paper presents a novel variant of the Broyden quasi-Newton secant-type method aimed at solving constrained mixed generalized equations, which can include functions that are not necessarily differentiable. The proposed method integrates the classical secant approach with techniques inspired by the Conditional Gradient method to handle constraints effectively. We establish local convergence results by applying the contraction mapping principle. Specifically, under assumptions of Lipschitz continuity, a modified Broyden update for derivative approximation, and the metric regularity property, we show that the algorithm generates a well-defined sequence that converges locally at a Q-linear rate.