Clarke Subdifferential in Nonsmooth Analysis

• Clarke subdifferential is a set-valued generalization of the derivative that uses convex hulls of limiting gradients to address nondifferentiability in Lipschitz functions.
• It is always nonempty, convex, and compact, and supports advanced calculus rules like sum and chain rules which enhance analysis in variational problems.
• Applications span spectral geometry, dynamical systems regularization, and deep learning, making it essential for robust nonsmooth optimization techniques.

A Clarke subdifferential is a set-valued generalization of the derivative that captures the limiting behavior of gradients of locally Lipschitz functions—particularly at points of nondifferentiability—through convex hulls of limiting gradients. This construct plays a foundational role in nonsmooth analysis, optimization, variational problems, and dynamical systems regularization. Its mathematical structure, calculus rules, and characterizations connect it to classical convex subdifferentials, generalized gradients, and Filippov regularization, with applications spanning spectral geometry, Banach space theory, and automatic differentiation.

1. Formal Definition and Fundamental Properties

Let $f:\mathbb{R}^n \to \mathbb{R}$ be a Lipschitz function. By Rademacher's theorem, $f$ is differentiable almost everywhere. The Clarke subdifferential at $x$, denoted $\partial^C f(x)$, is defined by

$$\partial^C f(x) := \operatorname{co}\left\{ \lim_{k\to\infty} \nabla f(x_k) : x_k \to x,\, x_k \text{ differentiability points of } f \right\},$$

where $\operatorname{co}$ denotes the convex hull. Equivalently, it is the intersection of closed convex hulls of gradients in every neighborhood of $x$: $$\partial^C f(x) = \bigcap_{\varepsilon>0}\operatorname{co}\left\{\nabla f(y): y\in B_\varepsilon(x)\setminus N_f \right\}$$ with $N_f$ the null set where $f$ is not differentiable (Bivas et al., 2020).

Key properties:

• $\partial^C f(x)$ is always nonempty, convex, and compact.
• The mapping $x \mapsto \partial^C f(x)$ is upper semicontinuous in the sense of set-valued maps.
• If $f$ is differentiable at $x$, then $\partial^C f(x) = \{ \nabla f(x) \}$.
• The Clarke subdifferential for a Banach space setting is defined analogously via limiting subgradients and convexification (Daniilidis et al., 2018, Nam et al., 2013).

2. Geometric and Analytic Structure

The geometry of the Clarke subdifferential can be understood as follows:

• It collects all "limiting slopes" of $f$ at points arbitrarily close to $x$, convexifying these limits.
• For functions with maximal Clarke subdifferential (termed "Clarke-saturated"), the subdifferential at every point fills the largest possible convex, compact set permitted by the Lipschitz constant (the closed dual ball) (Daniilidis et al., 2018).
• In the context of stratifiable or semi-algebraic functions, the Clarke subdifferential admits representations combining limits of gradients along strata, vertical cones, and normal cones to the domain (Drusvyatskiy et al., 2012, Lewis et al., 2021).
• The subdifferential for directionally Lipschitzian stratifiable functions is generated by the convex hull of limiting gradients, the cone of scaled limiting gradients (for vertical directions), and the convex normal cone to the domain.

3. Calculus, Chain Rule, and Variational Formulae

Clarke subdifferential calculus comprises sum, product, and chain rules, as well as mean-value and Fermat type results:

• Sum rule: $$\partial^C(f+g)(x) \subset \partial^C f(x) + \partial^C g(x)$$, with equality under additional regularity (e.g., if one summand is Clarke-regular) (Nam et al., 2013).
• Chain rule: For $g:\mathbb{R}^m\to\mathbb{R}$ Lipschitz and $H:\mathbb{R}^n\to\mathbb{R}^m$ continuously differentiable,

$$\partial^C(g\circ H)(x) \subset (\nabla H(x))^\top \partial^C g(H(x))$$

with equality under openness/local surjectivity or regularity (Guan et al., 2024).

• For composite models (e.g., matrix factorization, factorization machines), exact Clarke subdifferential chain rules hold in the overparameterized regime, tied to the local surjectivity of the associated multilinear map (Guan et al., 2024).
• Directional derivative: The Clarke generalized directional derivative,

$$f^{\circ}(x; v) = \limsup_{y\to x, t\downarrow 0} \frac{f(y + t v) - f(y)}{t},$$

leads to the dual characterization:

$$\partial^C f(x) = \{ x^* \mid \langle x^*, v \rangle \leq f^{\circ}(x; v) \ \forall v \}.$$

• Fermat rule: If $f$ attains a local minimum at $x$, then $0 \in \partial^C f(x)$.
• Mean value theorem: For Lipschitz $f$, any two points are joined by an intermediate point whose Clarke subdifferential realizes the increment by evaluation (Voloshyn, 2016).

4. Filippov Regularization and Characterization of Clarke Subdifferentials

A central result in the theory identifies which set-valued maps arise as Clarke subdifferentials of Lipschitz functions (Bivas et al., 2020):

• Given a cusco (convex, compact, upper semicontinuous) map $\Phi$, $\Phi$ is a Clarke subdifferential if and only if
  1. $\Phi$ is Filippov-representable: $\Phi = F_g$ for some measurable selection $g$ (Filippov regularization);
  2. $g$ satisfies the nonsmooth Poincaré symmetry: $\partial_i g_j = \partial_j g_i$ in the sense of distributions.

The mapping $x \mapsto \partial^C f(x)$ is exactly the minimal Filippov envelope among such cusco maps, and Clarke subdifferentials can be tested via local convex hull stability $\Phi = m(\Phi)$. This framework unifies the subdifferential calculus with regularization theory for discontinuous ODEs and differential inclusions.

5. Connections to Other Subdifferential Theories

The Clarke subdifferential is closely related to the Mordukhovich (basic/limiting) and convex subdifferentials:

• In locally Lipschitz settings, Clarke’s subdifferential is the closed convex hull of the Mordukhovich subdifferential:

$$\partial^C f(x) = \operatorname{cl}^* \operatorname{co} \partial^M f(x)$$

where $\partial^M$ is the limiting subdifferential (Nam et al., 2013).

• For convex $f$, the Clarke, limiting/Mordukhovich, and Fenchel subdifferentials coincide.
• For directionally Lipschitzian, stratifiable, or semi-algebraic functions, Clarke subdifferential formulas integrate limiting gradients over strata and normal cones to domains (Drusvyatskiy et al., 2012, Lewis et al., 2021).
• For marginal functions and optimization over set-valued mappings, Clarke’s coderivative calculus yields explicit subdifferential estimates (Bouza et al., 2021).

6. Maximality, Lineability, and Pathological Examples

The set of Lipschitz functions with maximal Clarke subdifferential (i.e., for which $\partial^C f(x)$ fills the entire closed dual ball everywhere) is unexpectedly large. Daniilidis–Flores have shown that:

• The collection of "Clarke-saturated" functions is lineable and contains closed subspaces isometric to $\ell^\infty(\mathbb{N})$; in fact, the set is spaceable in the Lipschitz norm topology (Daniilidis et al., 2018).
• Explicit constructions, based on measurable splitting of $\mathbb{R}$, yield uncountable-dimensional subspaces of such saturated functions.
• Most Lipschitz functions (Baire category sense) are saturated, but the explicit constructive proof situates these objects firmly within standard function spaces.

Pathologies: Such functions are "critical everywhere" for nonsmooth optimization, complicating genericity arguments for regularity or isolated minimizers.

7. Applications, Variational Problems, and Broader Context

Clarke subdifferential methods are fundamental in:

• Spectral variational problems: They underlie Euler–Lagrange characterizations of critical metrics for spectral functionals, such as Laplace and Steklov eigenvalues, enabling precise computation of subdifferentials for eigenvalue maps and their combinations under smooth and nonsmooth perturbations (Petrides et al., 2024).
• Optimization theory: Clarke’s calculus allows robust analysis of composite and marginal functions, with precise upper estimates via coderivative techniques that do not rely on Asplund space assumptions (Bouza et al., 2021).
• Automatic differentiation and nonsmooth dynamics: Conservative fields in deep learning correspond to selections from Clarke subdifferentials plus normal cones to strata in a Whitney stratification, governing chain rules in nonsmooth objectives (Lewis et al., 2021).
• General vector spaces: Modified Clarke subdifferentials extend the calculus to algebraically Lipschitz maps with respect to Minkowski functionals, accommodating broader contexts beyond normed spaces (Voloshyn, 2016).

The versatility of the Clarke subdifferential makes it essential for nonsmooth analysis, unifying classical subdifferential, generalized gradient, and regularization approaches, with applications from dynamical systems to large-scale variational geometry.

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Key References:

• "Characterization of Filippov representable maps and Clarke subdifferentials" (Bivas et al., 2020)
• "Linear structure of functions with maximal Clarke subdifferential" (Daniilidis et al., 2018)
• "The structure of conservative gradient fields" (Lewis et al., 2021)
• "A Unified Approach to Convex and Convexified Generalized Differentiation of Nonsmooth Functions and Set-Valued Mappings" (Nam et al., 2013)
• "Algebraic Lipschitz and Subdifferential Calculus in General Vector Spaces" (Voloshyn, 2016)
• "On Clarke's Subdifferential of Marginal Functions" (Bouza et al., 2021)
• "Clarke subgradients for directionally Lipschitzian stratifiable functions" (Drusvyatskiy et al., 2012)
• "Critical metrics of eigenvalue functionals via Clarke subdifferential" (Petrides et al., 2024)
• "On subdifferential chain rule of matrix factorization and beyond" (Guan et al., 2024)