Coderivative-Based Metric Regularity

• The paper presents coderivative-based criteria for metric regularity, defining a Lipschitz-type invertibility property through dual-space estimates.
• It utilizes limiting normal cones and the Mordukhovich coderivative to derive both qualitative and quantitative tests for set-valued mappings.
• The framework extends to directional and Hölder regularity, providing computable conditions for non-smooth and infinite-dimensional optimization problems.

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 (Ioffe, 2015).

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, $$\mathrm{reg}\,F(\bar{x}|\bar{y}) = \limsup_{(u,v)\to(\bar{x},\bar{y})}\|\big(D^*F(u,v)\big)^{-1}\|^+$$ (Ioffe, 2015).

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 $\operatorname{Graph} F$ (Ioffe, 2015).

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 $y \in F(x) + B(\bar{y}',\delta)$ (Ngai et al., 2013).
• 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 $\leq 1/m$ (Ngai et al., 2013).

• 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 (Cuong, 2023, Huynh et al., 2015).

• 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 $(u,v)$ (Huynh et al., 2015).

4. Quantitative Rate Formulas and Computational Illustrations

Quantitative regularity rates are expressed by explicit coderivative estimates:

$$\operatorname{sur} F(\bar{x}|\bar{y}) = \lim_{\varepsilon\to 0} \inf \left\{ \|x^*\| : x^* \in D^*F(u,v)(y^*),~\|y^*\|=1,~\|(u-x, v-y)\|<\varepsilon \right\}$$

(Ioffe, 2015). 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 $x \in \operatorname{int} B(c,r)$ (Hien, 2024).
• 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 $i$ (Hien et al., 2024).

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 (Ioffe, 2015).

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 (Durea et al., 2012).
• 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 (Ngai et al., 2013, Huynh et al., 2015).
• 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 (Clason et al., 2015).

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 (Ioffe, 2015, Ngai et al., 2013, Huynh et al., 2015, Durea et al., 2012, Hien, 2024, Hien et al., 2024, Clason et al., 2015, Cuong, 2023).