Mordukhovich Limiting Subdifferentials

• Mordukhovich limiting subdifferentials are a fundamental concept defined via normal cones to the epigraph, extending classical differentiation to nonsmooth, nonconvex functions.
• They enable rigorous calculus rules such as the sum and chain rules under qualification conditions, enhancing analysis in variational and sensitivity contexts.
• Applications span nonsmooth optimization, coderivative analysis, and stability studies in both finite and infinite dimensional settings.

Mordukhovich limiting subdifferentials, also known as basic or limiting subdifferentials, form a foundational apparatus in variational analysis for characterizing and analyzing nonsmooth, nonconvex functions and set-valued mappings in both finite and infinite dimensional settings. They are closely linked to the theory of generalized differentiation developed by B. S. Mordukhovich and play a central role in subdifferential calculus, stability and sensitivity analysis, as well as optimality conditions in nonsmooth optimization.

1. Rigorous Definition and Elementary Properties

For an extended-real-valued function $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$, with $\bar{x} \in \operatorname{dom} f$, the Mordukhovich (limiting) subdifferential at $\bar{x}$ is defined via the normal cone to the epigraph of $f$ by

$$\partial f(\bar{x}) := \left\{ x^* \in \mathbb{R}^n \;\middle|\; (x^*,-1) \in N_{\rm lim}\left((\bar{x},f(\bar{x})); \mathrm{epi}\,f\right) \right\},$$

where the limiting normal cone $N_{\rm lim}$ to a closed set $\Omega \subset \mathbb{R}^n$ at $\bar{x}$ is

$$N_{\rm lim}(\bar{x}; \Omega) := \left\{ v \in \mathbb{R}^n \;\middle|\; \exists \, x_k \to \bar{x},\; v_k \to v,\; v_k \in \widehat{N}(x_k; \Omega) \right\},$$

with $\widehat{N}(x;\Omega)$ the (Fréchet) regular normal cone, defined as

$$\widehat{N}(x; \Omega) := \left\{ v \in \mathbb{R}^n \;\middle|\; \limsup_{y \to x,\, y \in \Omega} \frac{\langle v, y-x \rangle}{\|y-x\|} \leq 0 \right\}.$$

Equivalent formulations include the “limits of gradients” when $f$ is locally Lipschitz, i.e.,

$$\partial_L f(x) = \left\{ p \in \mathbb{R}^n : \exists (x_k) \to x,\ x_k \in D,\ \nabla f(x_k) \to p \right\},$$

with $D$ the set of differentiability points of $f$ (Daniilidis et al., 2024).

Key features:

• $\partial f(\bar{x})$ is always closed, but not necessarily convex. Its convex hull recovers the Clarke (convexified) subdifferential.
• For $f$ convex, $\partial f(\bar{x})$ coincides with the classical convex subdifferential.
• For $f \in C^1$ near $\bar{x}$, $\partial f(\bar{x}) = \{ \nabla f(\bar{x}) \}$.
• For indicator functions, the subdifferential corresponds to the normal cone (Benko et al., 2017, An et al., 2024).

2. Calculus Rules and Qualification Conditions

The calculus of limiting subdifferentials mirrors subdifferential rules in convex analysis, but essential differences arise due to nonconvexity. Principal rules include:

• Sum Rule: For $f_1, f_2$ lsc, finite at $\bar{x}$,

$$\partial(f_1 + f_2)(\bar{x}) \subset \partial f_1(\bar{x}) + \partial f_2(\bar{x}),$$

with equality if one is locally Lipschitz at $\bar{x}$, provided the qualification condition $$0 \notin \partial^\infty f_1(\bar{x}) + \partial^\infty f_2(\bar{x})$$ holds, where $\partial^\infty$ denotes the singular subdifferential (Benko et al., 2017).

• Chain Rule: For $F: \mathbb{R}^n \to \mathbb{R}^m$ strictly differentiable at $\bar{x}$, $g: \mathbb{R}^m \to \mathbb{R} \cup \{+\infty\}$ lsc at $F(\bar{x})$,

$$\partial(g \circ F)(\bar{x}) \subset \nabla F(\bar{x})^T \partial g(F(\bar{x}))$$

under a no-singularity intersection with $\ker \nabla F(\bar{x})^T$; equality if $g$ is locally Lipschitz (Benko et al., 2017).

• Scalar Multiple: For $\alpha > 0$, $$\partial(\alpha f)(\bar{x}) = \alpha \partial f(\bar{x})$$ and $$\partial^\infty(\alpha f)(\bar{x}) = \alpha \partial^\infty f(\bar{x})$$.
• Product Rule: If one function is $C^1$ near $\bar{x}$, an explicit product rule applies under a no-singularity condition (Benko et al., 2017).
• Second-Order Subdifferential: For $f$ prox-regular at $(\bar{x},\bar{v})$,

$$\partial^2 f(\bar{x},\bar{v})(u) := D^* (\partial f)(\bar{x},\bar{v})(u) = \{ w \mid (w,-u) \in N((\bar{x},\bar{v}); \operatorname{gph} \partial f) \}$$

(An et al., 2024).

3. Directional and Set-Based Generalizations

Directional limiting subdifferentials refine sensitivity analysis by restricting sequences converging to $\bar{x}$ along a prescribed direction $u$. The directional limiting normal cone is defined as

$$N_{\rm lim}(\bar{x}; u; \Omega) := \left\{ v : \exists t_k \downarrow 0,\ u_k \to u,\ x_k=\bar{x}+t_k u_k \in \Omega,\ v_k \in \widehat{N}(x_k; \Omega),\ v_k \to v \right\}$$

and the directional subdifferential analogously via the epigraph (Benko et al., 2017). Such constructions capture finer geometric information, especially important in stability and sensitivity analysis.

Generalized forms with respect to a closed set $\mathcal{C}$ are formulated as

$$\partial_\mathcal{C} f(x) := \{ x^* \mid (x^*,-1) \in N_\mathcal{C}((x,f(x)); \operatorname{epi} f) \},$$

with $N_\mathcal{C}$ the limiting normal cone relative to $\mathcal{C}$, enabling subdifferential calculus on constraint manifolds and in variational geometries (Qin et al., 2023).

4. Analytical and Geometric Pathologies

Recent results have demonstrated the geometric complexity of limiting subdifferentials, even for everywhere differentiable Lipschitz functions. For any nonempty compact convex set $K \subset \mathbb{R}^n$ with nonempty interior, there exists a differentiable locally Lipschitz function $f$ so that $\partial_L f(\bar{x}) = K$ at some $\bar{x}$ (Daniilidis et al., 2024). This reveals a substantial gap between $C^1$ and merely differentiable functions in terms of subdifferential geometry, with upper semicontinuity and closedness being essentially the only universal regularity properties in the absence of higher smoothness.

5. Optimality and Stability: Necessary and Sufficient Conditions

The Mordukhovich subdifferential underpins necessary optimality conditions for minimization problems, both unconstrained and constrained. For $u^*$ a minimizer in $L^s(\Omega)$ for

$$\min_{u \in U_{\mathrm{ad}}} f(u) + \int_\Omega |u|^p,\quad p \in [0,1),$$

Fermat-type (stationarity) conditions read

$$0 \in f'(u^*) + \partial q_{s,p}(u^*) + N_{U_{\mathrm{ad}}}(u^*),$$

where $N_{U_{\mathrm{ad}}}(u^*)$ is the limiting normal cone to $U_{\mathrm{ad}}$ at $u^*$ (Mehlitz et al., 2021).

Second-order optimality is characterized by the second-order limiting subdifferential of the Lagrangian. For a $C^{1,1}$-smooth constrained optimization problem, necessary and sufficient second-order conditions are formulated via the second-order limiting subdifferential

$\partial^2 L(\bar{x}, \lambda, \mu)(v)$

in terms of directional positivity in the critical cone, extending the classical results to nonsmooth, nonconvex settings (An et al., 2024).

6. Applications and Representativity

The calculus of Mordukhovich subdifferentials, including coderivatives for multifunctions, is crucial in modern variational analysis—governing algorithms for optimization under nonsmooth or nonconvex settings, robust sensitivity theory, coderivative criteria for properties such as calmness and Lipschitz-like (Aubin) regularity, and the formulation of necessary optimality conditions in infinite-dimensional control and learning problems (Benko et al., 2017, Qin et al., 2023, An et al., 2024, Mehlitz et al., 2021).

Illustrative examples highlight the mechanisms and subtlety of the theory:

• For $f(x) = |x| + \sqrt{|x|}$, the sum rule’s conclusion $\partial f(0) = [-1,1]$ directly reflects the sum of subdifferentials with the qualification condition satisfied (Benko et al., 2017).
• For functionals on Lebesgue spaces with sparsity-promoting terms ($\ell_p$ with $p \in [0,1)$), explicit formulas for both regular and limiting/ singular subdifferentials illuminate the interplay of nonconvexity, non-Lipschitzianity, and optimality structure (Mehlitz et al., 2021).

7. Summary Table: Subdifferential Types and Defining Properties

Subdifferential Type	Definition	Key Properties/Context
Fréchet (regular) subdifferential $\widehat\partial f(x)$	Regular normals to $\operatorname{epi} f$ or supporting inequalities	Local support, closed, can be empty
Limiting (Mordukhovich) subdifferential $\partial f(x)$	Limits (in pairs) of Fréchet normals/subgradients	Always closed, not necessarily convex or single-valued
Clarke subdifferential $\partial_C f(x)$	Convex hull of limiting subdifferential	Convexified, captures all generalized directions

The Mordukhovich limiting subdifferential thus generalizes classical differentiation notions, bridging smooth, convex, and fully nonconvex settings, and provides the analytical machinery essential for advanced variational analysis, optimality, and stability in nonsmooth optimization and control (Benko et al., 2017, An et al., 2024, Daniilidis et al., 2024, Mehlitz et al., 2021, Qin et al., 2023, Drusvyatskiy et al., 2012).