Generalized Newton Method

• Generalized Newton Method is a class of algorithms that extend classical Newton’s method using graphical derivatives to linearize nonsmooth mappings.
• It leverages metric regularity and coderivative conditions to ensure robust, often superlinear, local and global convergence in complex variational problems.
• The method broadens the applicability beyond semismooth approaches, enabling practical solutions for nonsmooth, set-valued equations and variational inequalities.

The Generalized Newton Method comprises a class of algorithms that extend the classical Newton’s method to broad families of nonsmooth and set-valued equations, including generalized equations and inclusions defined by variational analysis. These methods are designed for problems where the underlying mapping may be nonsmooth, non-Fréchet differentiable, or even set-valued, and in which classical smooth calculus does not suffice. The principal innovation is to employ generalized differential tools—such as graphical derivatives, coderivatives, or related generalized Jacobians—to construct Newton-type iterations with strong local (often superlinear) and, in favorable cases, global convergence guarantees, frequently under metric regularity hypotheses.

1. Graphical Derivative-Based Newton Algorithm

The core problem is to solve nonsmooth equations $H(x) = 0$ for continuous, but potentially nonsmooth, mappings $H: \mathbb{R}^n \to \mathbb{R}^n$. The fundamental step in the generalized Newton method based on graphical derivatives is to define the search direction $d^k$ at each iterate $x^k$ as a solution to the inclusion

$-H(x^k) \in D H(x^k) (d^k),$

where $D H(x^k)$ denotes the graphical derivative of $H$ at $x^k$. In the smooth case, this reduces to solving $-H(x^k) = H'(x^k) d^k$, the classical Newton step. For nonsmooth $H$, the graphical derivative $DH(x)$—which captures the directional limiting behavior of the mapping—allows for the linearization of $H$ in directions where classical derivatives are undefined.

The numerical iteration proceeds as:

$x^{k+1} = x^k + d^k$

with $d^k$ determined as above. Well-posedness (existence of $d^k$) and local convergence require $H$ to be metrically regular at a solution—this allows the graphical derivative inclusion to be solvable and the iteration to progress.

2. Role of Metric Regularity and Uniform Error Estimates

Metric regularity serves as the cornerstone for the convergence analysis of the graphical-derivative-based Newton scheme. A mapping $H$ is metrically regular at $x^*$ for $0$ with modulus $\mu > 0$ if, locally,

$$\text{dist}(x, H^{-1}(0)) \leq \mu \| H(x) \| \quad \text{for } x \text{ near } x^*.$$

This property is characterized via coderivatives, specifically, by the invertibility condition of the graphical derivative at $x^*$. The metric regularity hypothesis ensures not only well-posedness of the Newton subproblem but also enables the derivation of crucial uniform error bounds.

A central result is that, under metric regularity with modulus $\mu$ and for iterates sufficiently close to the solution $x^*$,

$$\| x^{k+1} - x^* \| \leq \frac{\mu}{1 - \mu} \| x^k - x^{k-1} \|.$$

This error estimate, analogous to those in certain proximal-point schemes [Artacho and Goeffroy], quantifies the contraction and robustness of the iteration—even permitting inexact computation of the direction $d^k$ if the errors are controlled relative to $\mu$.

3. Variational Analysis and Generalized Differentiation in Algorithm Construction

The development and analysis of the generalized Newton method leverage advanced tools from variational analysis:

• Graphical derivatives ($DH$): Capture the tangent cone to the graph of $H$ at a point and generalize the Jacobian for nonsmooth mappings.
• Coderivatives: Provide the adjoint to the graphical derivative, essential for analyzing metric regularity and error bounds.
• Generalized Jacobians and limiting subdifferentials: Supply the calculus for constructing and analyzing the iteration in the absence of smoothness.

Specifically, the coderivative-based characterization of metric regularity (involving surjectivity-type or invertibility-type conditions) underlies both the existence theory for the Newton subproblem and the convergence rate guarantees.

4. Comparison with Semismooth and $B$-Differentiable Newton Methods

Traditional Newton-type schemes for nonsmooth problems—such as semismooth and $B$-differentiable Newton methods—typically restrict their application to Lipschitzian equations satisfying semismoothness or $B$-differentiability. These rely on the generalized Jacobian in the sense of Clarke or on some special directional differentiability condition.

The graphical derivative-based Newton method admits cases beyond the semismooth framework by not imposing any semismoothness assumptions on $H$. In situations where semismoothness fails (i.e., the nonlinearity exhibits nondirectional or irregular behavior), the method remains applicable, provided metric regularity holds. Thus, the approach properly contains the semismooth Newton method as a special case and extends applicability and robustness.

5. Global and Superlinear (Kantorovich-Type) Convergence

Beyond local behavior, the graphical derivative-based generalized Newton method admits a Kantorovich-type global convergence theorem: if the mapping $H$ is metrically regular and certain further continuity and regularity (assumptions (H1) and (H2) in the original presentation) are met, then the iteration converges globally from initial guesses sufficiently close to a solution with superlinear rate.

Superlinear convergence follows from the fact that, under these hypotheses, the error between iterates contracts faster than linearly:

$\| x^{k+1} - x^* \| = o( \| x^k - x^* \| ),$

assuming the graphical derivative mapping retains suitable regularity in a neighborhood.

6. Practical Implications and Robustness to Nonsmoothness

The method allows for practical robustness to the lack of classical differentiability:

• No requirement of semismoothness: Graphical derivatives support mappings exhibiting genuine nonsmoothness.
• Tolerance to inexact computation: Convergence is retained even if the Newton direction is computed approximately, provided the error is small relative to the metric regularity modulus.
• Applicability to nonsmooth equation classes: Inclusion problems, nonsmooth variational equalities, and non-Lipschitz mappings are encompassed if metric regularity and associated coderivative estimates hold.

This broadens the relevance of Newton-type methods to settings previously inaccessible to standard or semismooth Newton methods, with significant implications for variational inequalities, nonsmooth optimization, and operational models involving set-valued mappings.

7. Connections to Proximal Frameworks and Broader Variational Methodologies

There is a strong parallel between the graphical derivative-based Newton method and inexact proximal-point algorithms analyzed under metric regularity [Artacho and Goeffroy]. Both frameworks rely upon error estimates tied to metric regularity and employ variational analytic constructs (typically coderivatives) to ensure robustness in the face of nonsmoothness and computational inexactness. This conceptual alignment suggests that the convergence guarantees, error bounds, and regularity results obtained for the Newton-type method have immediate implications for proximal-type methods, and vice versa, reinforcing the centrality of metric regularity in modern variational numerical analysis.