Second-Order Gph-Regularity

• Second-order gph-regularity is a structural property for mappings that enables a parabolic expansion to derive precise second-order optimality conditions in variational analysis.
• It ensures no-gap conditions by aligning necessary and sufficient optimality criteria in nonsmooth, hierarchical, and bilevel programming contexts.
• The concept extends classical second-order analysis to complex, nonconvex problems, preserving outer second-order regularity under minimal assumptions.

Second-order gph-regularity is a structural property for set-valued and single-valued mappings in variational analysis and optimization. It plays a fundamental role in obtaining precise second-order necessary and sufficient optimality conditions in nonsmooth constrained problems, particularly beyond the field of smooth or convex sets. This conceptual framework allows the parabolic expansion of nonsmooth mappings and enables “no-gap” second-order theory—i.e., alignment of necessary and sufficient conditions without artificial restrictions on sets or functions—across a broad spectrum of hierarchical, bilevel, and nonconvex programming contexts.

1. Formal Definition and Basic Structure

Let $g : \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a locally Lipschitz and second-order directionally differentiable function at $x^*$. Second-order gph-regularity at $x^*$ in direction $d \in \mathbb{R}^n$ requires that for every path $w : \mathbb{R}_+ \rightarrow \mathbb{R}^n$ with $t w(t) \to 0$ as $t \downarrow 0$, there exists a remainder $r(t^2) = o(t^2)$ such that

$$g(x^* + t d + \tfrac{1}{2} t^2 w(t)) = g(x^*) + t g'(x^*; d) + \tfrac{1}{2} t^2 g''(x^*; d, w(t)) + r(t^2)$$

as $t \downarrow 0$. If this expansion holds for all $d \in \mathbb{R}^n$, $g$ is said to be second-order gph-regular at $x^*$ (Liu et al., 4 Nov 2025).

• The term “gph” refers to a graphical expansion that exactly matches this parabolic (second-order) model along all sufficiently tangential curves.
• Every $C^2$ mapping is second-order gph-regular.
• Second-order gph-regularity is strictly stronger than second-order epi-regularity, meaning it implies the latter, but not vice versa.

2. Key Properties and Equivalent Criteria

Second-order gph-regularity is preserved under a range of operations:

Construction	Property	Reference
Absolute value, $\ell_1$ and $\ell_2$ norms, $\min_i x_i$, projections	Each is second-order gph-regular	Proposition 2.9 (Liu et al., 4 Nov 2025)
Piecewise-$C^2$ (PC$^2$) mappings	Second-order gph-regular at all points	Proposition 2.11 (Liu et al., 4 Nov 2025)
Composition $f \circ g$	Gph-regular if $f$ and $g$ each are	Proposition 2.13 (Liu et al., 4 Nov 2025)
Local inverse mapping	Gph-regular if the Clarke–Jacobian is full rank	Proposition 2.14 (Liu et al., 4 Nov 2025)
Implicit function mapping	Gph-regular if the constraint has full rank	Corollary 2.15 (Liu et al., 4 Nov 2025)

Further, if $G : \mathbb{R}^n \rightarrow \mathbb{R}^m$ is second-order gph-regular at $x^*$ and a metric subregularity constraint qualification (MSCQ) holds, the feasible set $\Psi = \{ x : G(x) \in K \}$ inherits outer second-order regularity from $K$ at $G(x^*)$ (Proposition 3.6 (Liu et al., 4 Nov 2025)).

3. Role in Variational and Second-Order Analysis

Second-order gph-regularity is tailored to the parabolic-curve approach to variational optimality conditions. Consider the constrained program

$\min f(x) \quad \text{subject to } G(x) \in K,$

with $f, G$ locally Lipschitz and second-order directionally differentiable, $G$ gph-regular, and $K \subset \mathbb{R}^m$ a closed set.

For parabolic sequences $x^k = x^* + t_k d + \frac{1}{2} t_k^2 w^k$, $t_k \downarrow 0$, $t_k w^k \to 0$, $G$ admits the expansion

$$G(x^k) = G(x^*) + t_k G'(x^*; d) + \frac{1}{2} t_k^2 G''(x^*; d, w^k) + o(t_k^2),$$

and outer second-order regularity of $K$ at $G(x^*)$ in direction $G'(x^*; d)$ ensures that

$$\operatorname{dist}(G''(x^*; d, w^k), T^2_K(G(x^*); G'(x^*; d))) \rightarrow 0,$$

where $T_K$ and $T^2_K$ denote the tangent and second-order tangent sets. MSCQ enables the same behavior at the level of feasible points in $x$, i.e., $\operatorname{dist}(w^k, T^2_{\Psi}(x^*; d)) \to 0$, permitting direct passage to second-order necessary and sufficient conditions.

4. Second-Order Optimality Conditions and "No-Gap" Theorems

Second-order gph-regularity is essential for formulating "no-gap" necessary and sufficient optimality conditions in nonconvex and nonsmooth optimization. The following results hold under gph-regularity and MSCQ (Liu et al., 4 Nov 2025):

• First-order necessary condition:

$$0 \in \operatorname{argmin}\{ f'(x^*; d) : G'(x^*; d) \in T_K(G(x^*)) \}$$

• Second-order necessary condition: For any $d$ with $G'(x^*; d) \in T_K(G(x^*))$, $f'(x^*; d) \leq 0$, and any $w$ with $G''(x^*; d, w) \in T^2_K(G(x^*); G'(x^*; d))$,

$f''(x^*; d, w) \geq 0$

• Second-order sufficient condition: If, in addition, outer second-order regularity holds for the feasible set and for all nonzero $d$ with $f'(x^*; d) = 0$,

$\min_{w \in T^2_\Psi(x^*; d)} f''(x^*; d, w) > 0,$

then $x^*$ satisfies a quadratic growth condition.

In the $C^2$ setting, the above recovers classical second-order conditions involving the Hessian of $f$ and the Lagrangian, but second-order gph-regularity enables precise analogues in fully nonsmooth regimes.

5. Applications in Bilevel and Hierarchical Optimization

Second-order gph-regularity determines the structural regularity required to obtain second-order necessary and sufficient conditions in bilevel and hierarchical optimization (Liu et al., 4 Nov 2025). For bilevel problems of the form

$$\min_{x, y} F(x, y) \quad \text{s.t. } H(x, y) = 0, \; G(x, y) \leq 0, \; y \in S(x)$$

where $$S(x) = \arg\min_y \{ f(x, y) : h(x, y) = 0,\, g(x, y) \leq 0 \}$$,

• If the lower-level problem satisfies Mangasarian-Fromovitz CQ, strong second-order sufficient condition (SSOSC), and constant rank CQ, the local solution mapping $y(x)$ is second-order gph-regular.
• This enables the implicit reformulation $y = y(x)$, with first and second-order necessary and sufficient conditions derivable without recourse to higher complexity, even in the absence of uniqueness of so-called “lower-level multipliers.”
• If the lower-level LICQ holds, all second-order conditions can be expressed solely in terms of the quadratic blocks of the data $$(\nabla^2_{xx} F, \nabla^2_{xy} F, \nabla^2_{yy} f,\ldots)$$ and constraints, without reference to higher-order expansions of the optimal value function.

The gph-regularity property enables a complete second-order sensitivity and optimality theory in such nonconvex and nonsmooth settings.

6. Relationship to Other Regularity and Second-Order Notions

Second-order gph-regularity (in the sense of Liu et al. (Liu et al., 4 Nov 2025)) sits between classical $C^2$ regularity and the weaker notion of second-order epi-regularity:

• All $C^2$ mappings are second-order gph-regular.
• Piecewise $C^2$, absolute value functions, minimization operators, and metric projections satisfy second-order gph-regularity.
• Second-order epi-regularity (epi-expansion along parabolic curves for epigraphs) is implied but not equivalent.

Importantly, second-order gph-regularity is strictly necessary for the preservation of outer second-order regularity under set pullbacks and for the validity of parabolic-type expansions of feasible sets and solution mappings—properties not shared by mere directional second-derivative existence.

Compared with the framework of tilt stability and generalized Hessians in variational analysis (see, e.g., (Drusvyatskiy et al., 2013)), gph-regularity is a property of the mapping or constraint itself. In contrast, tilt stability and metric regularity, via the positive-definite generalized Hessian, are properties of the subdifferential at a minimizer; these are vital for stability, quadratic growth, and equivalence of strong regularity in composite optimization problems, but are not formulated in terms of explicit graphical expansions.

7. Perspectives and Extensions

Second-order gph-regularity facilitates the extension of classical second-order analysis tools to broad nonsmooth, nonconvex, and hierarchical contexts:

• Outer second-order regularity of feasible sets and solution mappings can be preserved under composition, inversion, and implicit mapping when gph-regularity holds and rank conditions are met.
• The no-gap theory implies that under gph-regularity and metric subregularity (without convexity or uniqueness of multipliers), full second-order necessary and sufficient optimality systems for constrained and bilevel optimization drop entirely to the level of (generalized) second derivatives of the data.
• The concept is applicable to nonsmooth programs with disjunctive, complementarity, or complex hierarchical structures, wherever outer second-order regularity of the feasible region is crucial.
• Open problems include characterizing the maximal class of mappings or sets admitting second-order gph-regularity and exploring conditions under which gph-regularity yields analyticity or higher-order (Gevrey–$s$) expansions—which has been addressed in stochastic homogenization for Poisson processes in a different sense (Duerinckx et al., 2022).

A plausible implication is that the broader adoption of second-order gph-regularity will systematically bridge the analytic gap between smooth and nonsmooth second-order variational analysis, under minimal structural assumptions on data and constraints.