2025/04/24 by Roubal, Tomáš, Valdman, Jan
#49J53 #65K15 #90C33 #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2504.17649
We extend the classical third-order Halley iteration to the setting of generalized equations of the form 0 ∈ f(x) + F(x), where \(f\colon X\longrightarrow Y\) is twice continuously Fréchet-differentiable on Banach spaces and \(F\colon X\tto Y\) is a set-valued mapping with closed graph. Building on predictor-corrector framework, our scheme first solves a partially linearized inclusion to produce a predictor \(uk+1\), then incorporates second-order information in a Halley-type corrector step to obtain \(xk+1\). Under metric regularity of the linearization at a reference solution and Hölder continuity of \(f''\), we prove that the iterates converge locally with order \(2+p\) (cubically when \(p=1\)). Moreover, by constructing a suitable scalar majorant function we derive semilocal Kantorovich-type conditions guaranteeing well-definedness and R-cubic convergence from an explicit neighbourhood of the initial guess. Numerical experiments-including one- and two-dimensional test problems confirm the theoretical convergence rates and illustrate the efficiency of the Josephy-Halley method compared to its Josephy-Newton counterpart.