Secant-type method with feasible inexact projection for solving constrained mixed generalized equations

Abstract

This thesis addresses the solution of a mixed generalized equation of the form Find x ∈ C such that f(x) + g(x) + F(x) ∋ 0, where f:Ω → Rn is continuously differentiable, g:Ω → Rn is continuous (but not necessarily differentiable), F:Ω ⇉ Rn is a set-valued mapping with closed, nonempty graph, Ω ⊆ Rn is open, and C ⊂ Ω is a closed convex set. To handle the non-smoothness of g and the complexity of the set-valued term F, this work introduces and analyzes two iterative methods based on partial linearizations and feasible inexact projections. The first is a secant-type method, which utilizes first- and second-order divided differences of g to approximate its local behavior. The second is a quasi-Newton method employing a Broyden update to approximate the Jacobian of f, thus avoiding its exact computation. The convergence analysis of both methods is conducted under assumptions of metric regularity for the associated linearized mappings. Theoretical results establish local convergence of the sequences generated by the algorithms. For the secant method, it is shown that if the projection errors vanish, the convergence becomes superlinear, and in the exact projection case, the rate becomes quadratic. For the quasi-Newton method, assuming bounded deterioration of the Jacobian approximations, the sequence converges q-linearly. The analysis is supported by variational tools such as metric regularity, strong metric regularity, linearization error bounds, the Contraction Principle Theorem and a key perturbed metric regularity theorem. These tools enable a rigorous treatment of the impact of approximation errors in both the projections and Jacobian estimates, ensuring robustness and stability of the proposed methods. Overall, this thesis contributes new algorithms and a complete convergence theory for solving nonsmooth and set-valued generalized equations with convex constraints. The methods presented extend classical Newton-type strategies to more general, practical contexts involving nonsmooth components, variational terms, and computational inexactness.