---
title: Mordukhovich Limiting Subdifferentials
url: https://www.emergentmind.com/topics/mordukhovich-limiting-subdifferentials
type: topic
---

# Mordukhovich Limiting Subdifferentials

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 $f$ [2405.09206].

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 [1712.04704, 2412.19557].

## 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 [1712.04704].

- **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 [1712.04704].

- **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 [1712.04704].

- **Second-Order Subdifferential**: For $f$ prox-regular at $(\bar{x},\bar{v})$,
  \[
  \partial^2 f(\bar{x},\bar{v})(u) := D^* (\partial f)(\bar{x},\bar{v})(u) = \{ w \mid (w,-u) \in N((\bar{x},\bar{v}); \operatorname{gph} \partial f) \}
  \]
  [2412.19557].

## 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 [1712.04704]. 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 [2307.15389].

## 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 $\bar{x}$ [2405.09206]. 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 $u^*$ [2107.09340].

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 [2412.19557].

## 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 [1712.04704, 2307.15389, 2412.19557, 2107.09340].

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 [1712.04704].
- 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 [2107.09340].

## 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 [1712.04704, 2412.19557, 2405.09206, 2107.09340, 2307.15389, 1204.5794].

Source: https://www.emergentmind.com/topics/mordukhovich-limiting-subdifferentials