On Josephy-Halley method for generalized equations

TL;DR

This work addresses solving generalized equations $0 \in f(x) + F(x)$ by extending Halley's third-order method to a predictor-corrector Josephy-Halley framework. The predictor solves a partially linearized inclusion and the Halley-type corrector incorporates second-order information, achieving local convergence of order $2+p$ (cubic when $p=1$) under metric regularity and Hölder continuity of $f''$. A semilocal Kantorovich-type analysis using a scalar majorant yields computable convergence radii and guarantees well-definedness and $R$-cubic convergence from realistic starting guesses. Numerical experiments in 1D and 2D confirm the theory and show the Josephy-Halley method can outperform Josephy-Newton in iteration efficiency while maintaining comparable overall cost. The results advance robust, high-order methods for generalized equations with set-valued terms, with potential extensions to nonsmooth settings and large-scale variational problems.

Abstract

We extend the classical third-order Halley iteration to the setting of generalized equations of the form \[ 0 \in f(x) + F(x), \] 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 \(u_{k+1}\), then incorporates second-order information in a Halley-type corrector step to obtain \(x_{k+1}\). 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.

Figures (3)

• Figure 1: Halley vs. Newton method in case (i) of Example \ref{['Ex2']}. Comparison of convergence of $e_k$ versus the $k$-th iteration.
• Figure 2: Comparison of convergence: Halley vs. Newton of Example \ref{['Ex2']} (i).
• Figure 3: Comparison of convergence: Josephy-Halley vs. Josephy-Newton of Example \ref{['Ex2']} (ii).

Theorems & Definitions (15)

• Definition 2.1
• Theorem 2.1
• Theorem 2.2
• Theorem 3.1
• Proof 1
• Proposition 3.1
• Proof 2
• Lemma 4.1
• Remark 4.1
• Theorem 4.1