---
title: Coderivative-Based Metric Regularity
url: https://www.emergentmind.com/topics/coderivative-based-characterization-of-metric-regularity
type: topic
---

# Coderivative-Based Metric Regularity

Coderivative-based characterization of metric regularity provides a fundamental tool for the analysis of set-valued mappings in variational analysis, optimization, and control theory. Metric regularity describes a robust Lipschitz-type invertibility property of a set-valued mapping near a reference point, and the coderivative—a dual-space generalized derivative—yields both qualitative and quantitative criteria for this property across broad settings including Banach and finite-dimensional spaces, Asplund spaces, and beyond. The dual-space coderivative tests complement and often strengthen primal (slope- or tangential-derivative) criteria, and their power is especially evident in non-smooth, infinite-dimensional, or composite-structured mappings.

## 1. Metric Regularity and the Dual Coderivative Principle

Metric regularity for a set-valued mapping $F:X\rightrightarrows Y$ at $(\bar{x},\bar{y})\in\operatorname{Graph} F$ is defined by the existence of constants $\kappa>0$ and $\varepsilon>0$ such that for all $u\in B(\bar{x},\varepsilon),\,v\in B(\bar{y},\varepsilon)$, the inverse-distance estimate holds:
$$
d(u, F^{-1}(v)) \leq \kappa\,d(v,F(u)).
$$
The smallest such $\kappa$ is $\mathrm{reg}\,F(\bar{x}|\bar{y})$. Its reciprocal, the modulus of surjection $\operatorname{sur} F(\bar{x}|\bar{y})$, quantifies the best-possible Lipschitz constant for a local inverse selection [1505.07920].

The coderivative, specifically the limiting Mordukhovich coderivative $D^*F(\bar{x},\bar{y}):Y^*\rightrightarrows X^*$, is central:
$$
D^*F(\bar{x},\bar{y})(y^*) := \{ x^* \in X^* : (x^*, -y^*) \in N_{\operatorname{Graph} F}(\bar{x},\bar{y}) \},
$$
where $N$ denotes the limiting normal cone to the graph.

The critical coderivative modulus is
$$
\|D^*F(\bar{x},\bar{y})\|^- = \inf\{ \|x^*\| \mid x^* \in D^*F(\bar{x},\bar{y})(y^*),~\|y^*\|=1 \},
$$
the dual Banach constant.

The main coderivative regularity criterion is:

- **General (non-Asplund):** $\operatorname{sur} F(\bar{x}|\bar{y}) \geq \liminf_{(u,v)\to(\bar{x},\bar{y})}\|D^*F(u,v)\|^-$.

- **Asplund spaces (exact):** $\operatorname{sur} F(\bar{x}|\bar{y}) = \liminf_{(u,v)\to(\bar{x},\bar{y})}\|D^*F(u,v)\|^-$. Equivalently, $\mathrm{reg}\,F(\bar{x}|\bar{y}) = \limsup_{(u,v)\to(\bar{x},\bar{y})}\|\big(D^*F(u,v)\big)^{-1}\|^+$ [1505.07920].

This result yields a dual-space, quantitative and, in Asplund spaces, exact criterion for metric regularity, directly linking local surjectivity to lower bounds on coderivative norms.

## 2. Limiting Normal Cones and Coderivatives: Technical Framework

The coderivative is constructed using the geometry of the limiting normal cone $N(Q,x)$ for a closed set $Q\subset X$ at $x\in Q$, defined as cluster points of Fréchet normals $N^\widehat(Q,x_k)$ for nearby sequences $(x_k)\to x$:
$$
N(Q,x) = \left\{ x^* \in X^* \mid \exists (x_k, x_k^*) \to (x,x^*),\, x_k^* \in N^\wedge(Q,x_k) \right\},
$$
with
$$
N^\wedge(Q,x) = \{ x^* \mid \langle x^*, x' - x \rangle \leq o(\|x'-x\|)~\forall x'\in Q \}.
$$
The Mordukhovich coderivative $D^*F(\bar{x},\bar{y})$ is derived from this construction for $\operatorname{Graph} F$ [1505.07920].

In finite-dimensional spaces or for mappings with polyhedral graphs, $D^*F$ can be computed by polyhedral geometry, and for convex processes $A: X \rightrightarrows Y$, $D^*A(0,0) = A^*$, mirroring classical adjoint mappings. This enables explicit computation of surjection moduli in numerous examples.

## 3. Coderivative Characterizations in Directional and Hölder Regularity

Beyond classic metric regularity, the coderivative framework extends to *directional metric regularity* and to *Hölder-type* moduli.

- **Directional metric regularity**: For $F:X\to Y$, $(\bar{x},\bar{y}) \in \operatorname{gph} F$, direction $\bar{y}'\in Y$, $F$ is directionally metrically regular at $(\bar{x}, \bar{y})$ in direction $\bar{y}'$ if the metric inequality holds locally for $y \in F(x) + B(\bar{y}',\delta)$ [1304.7748].

- **Critical coderivative criterion (directional)**: If $F$ is closed-valued, convex-valued, pseudo-Lipschitz near $(\bar{x},\bar{y})$ and there exists $m>0$ such that
$$
\liminf_{\substack{(x,y_1,y_2)\to (\bar{x}, \bar{y}, \bar{y}) \\ \delta \downarrow 0}} d_*\big(0, D^*G(x,y_1,y_2)(T(\bar{y}',\delta))\big) > m,
$$
where $G(x)=F(x)\times F(x)$ and $T(\bar{y}',\delta)$ encodes the direction, then $F$ is directionally metrically regular at $(\bar{x},\bar{y})$ in the direction $\bar{y}'$ with modulus $\leq 1/m$ [1304.7748].

- **Hölder metric regularity**: If $F:X\to Y$ is set-valued, the $q$-order regularity is characterized by a coderivative lower bound:
$$
q\,\|z-y\|^{q-1} d(0, D^*F(x,z)(y^*)) \geq \tau
$$
for suitably chosen $(x,z,y^*)$, extending the classical Lyusternik–Graves theorem to the Hölder setting [2311.16731, 1508.01930].

- **Sufficient condition (directional Hölder):** If the critical set
$$
(0,0) \not\in \operatorname{Cr}^\gamma F((\bar{x},\bar{y}),(u,v)),
$$
where $\operatorname{Cr}^\gamma$ is defined via coderivative sequences, then $F$ is directionally Hölder metrically regular of order $\gamma$ in direction $(u,v)$ [1508.01930].

## 4. Quantitative Rate Formulas and Computational Illustrations

Quantitative regularity rates are expressed by explicit coderivative estimates:
$$
\operatorname{sur} F(\bar{x}|\bar{y}) = \lim_{\varepsilon\to 0} \inf
\left\{ \|x^*\| : x^* \in D^*F(u,v)(y^*),~\|y^*\|=1,~\|(u-x, v-y)\|<\varepsilon
\right\}
$$
[1505.07920]. The tightness of these rates holds for closed convex processes ($\operatorname{sur} A(0|0) = C^*(A^*)$), polyhedral mappings ($\operatorname{sur}F(x|y) = \min \{\|x^*\|: x^*\in D^*F(x,y)(y^*), \|y^*\|=1\}$), and subdifferential mappings ($\operatorname{sur}\,\partial f(x|f(x)) = \inf \{\|x^*\|: x^*\in \partial^2 f(x)(y^*),~\|y^*\|=1\}$).

For metric projections onto closed balls in Hilbert spaces, $P_{B(c,r)}:H\to B(c,r)$, the coderivative-based test gives:
- $P_{B(c,r)}$ is metrically regular at $x$ if and only if $x \in \operatorname{int} B(c,r)$ [2406.18377].
- For projections onto the positive cone $C\subset \mathbb{R}^n$, $P_C$ is metrically regular at $x$ if and only if $x_i\neq 0$ for all $i$ [2407.08007].

These explicit models demonstrate how coderivative analysis provides necessary and sufficient regularity conditions, reducible to geometric or linear-algebraic criteria.

## 5. Comparison with Primal and Tangential Criteria

The coderivative subsumes and extends classic slope-based and graphical-derivative (tangential) tests:

| Criterion        | Workspace                  | Key Modulus                                                |
|------------------|---------------------------|------------------------------------------------------------|
| Slope-based      | Primal/metric             | $\displaystyle \liminf \{-\inf_{\|h\|=1} d\phi(x,y;(h,\cdot))\}$ |
| Tangential       | Contingent/graphical      | $\displaystyle \liminf~C(D F(u,v))$                        |
| Coderivative     | Dual (normal/coderivative)| $\displaystyle \liminf~C^*(D^*F(u,v))$                     |

Always, $C(D F) \leq C^*(D^*F)$, so the coderivative estimate is never weaker and often sharper, especially in non-smooth contexts or when directional information is essential [1505.07920].

## 6. Extensions: Compositions, Perturbations, and Applications

Coderivative frameworks robustly extend to composite mappings and under perturbations:
- For $H(x) = G(F_1(x), F_2(x))$ with closed-graph $F_1, F_2, G$: coderivative conditions characterize metric regularity of $H$ up to alliedness of the graphs and a sum-of-coderivative lower bound [1210.4107].
- Under small Lipschitz perturbations ($g:X\to Y$), coderivative moduli control the degradation of regularity modulus for $F+g$, ensuring stability provided the Lipschitz constant is sufficiently small [1304.7748, 1508.01930].
- In infinite-dimensional/practical settings (e.g., Hilbert-space saddle-point mappings in PDE-constrained optimization), explicit pointwise coderivative formulas yield verifiable and computable tests for stability of saddle points and parameter identifiability [1509.06582].

## 7. Significance and Broader Impact

Coderivative-based characterization of metric regularity has unified and extended classical and modern regularity theory, enabling precise analysis in variational, control, optimization, and nonsmooth settings. Its strengths include:
- Clean, dual-space formulation amenable to non-smooth and infinite-dimensional analysis.
- Exact formulas in Asplund, convex, and polyhedral settings; tight quantitative rate bounds.
- Power in composite and structured applications, including stability and sensitivity analysis in parameterized and PDE-constrained optimization.
- Robustness to perturbations and strong connection to practical error bounds and algorithmic convergence.

Its fundamental role and explicit computability distinguish coderivative analysis as a principal tool in contemporary variational analysis and optimization theory [1505.07920, 1304.7748, 1508.01930, 1210.4107, 2406.18377, 2407.08007, 1509.06582, 2311.16731].

Source: https://www.emergentmind.com/topics/coderivative-based-characterization-of-metric-regularity