Second-Order Tangent Sets

• Second-Order Tangent Sets are generalized geometric constructions that capture parabolic approximations of closed sets, aiding in second-order conditions and set-valued differentiation.
• They include variants like the Bouligand outer and Ursescu inner constructions, each with distinct continuity and conicity characteristics, well-suited for both convex and nonconvex analysis.
• Their advanced calculus under metric subregularity enables precise second-order optimality conditions in complex optimization problems, even in infinite-dimensional or nonregular settings.

A second-order tangent set is a generalized geometric construction designed to encode parabolic (second-order) approximations of closed sets, manifolds, or constraint loci. They play a pivotal role in variational analysis, nonlinear optimization, and the theory of set-valued differentiation, as they provide the essential framework for second-order conditions and calculus beyond the scope of regular (first-order) tangent cones. Numerous variants of second-order tangent sets have emerged, including the classic Bouligand (outer) and Ursescu (inner) constructions, parabolic sets for complementarity structures, nonstandard-analysis-based formulations, and asymptotic cones as in Penot’s general theory. Recent advances have focused on refining their analytic and topological properties and adapting these concepts to nonconvex, nonregular, or infinite-dimensional settings.

1. Canonical Definitions and Notation

For a closed set $S \subset \mathbb{R}^n$, a reference point $\bar{x} \in S$, and a direction $d \in T_S(\bar{x})$ (where $T_S(\bar{x})$ is the Bouligand contingent tangent cone), two principal constructions are established:

• Outer (Bouligand) Second-Order Tangent Set:

$$T^2_S(\bar{x}; d) := \Big\{ w \in \mathbb{R}^n : \exists\, t_k \downarrow 0,\, w_k \to w,\; \bar{x} + t_k d + \tfrac{1}{2} t_k^2 w_k \in S \;\forall k \Big\}$$

This set collects all limiting parabolic “lifts” of secant vectors.

• Inner (Parabolic or Ursescu) Second-Order Tangent Set:

$$T_{\text{inner}}^2(S, \bar{x}, d) := \Big\{ w \in \mathbb{R}^n : \mathrm{dist}( \bar{x} + t d + \tfrac{1}{2} t^2 w, S ) = o(t^2)\ \text{as}\ t\downarrow 0 \Big\}$$

The inner variant is typically smaller, requiring higher-order proximity.

Additionally, the asymptotic second-order tangent cone (Penot) is defined as

$$\widetilde{T}^2_S(\bar{x}; d) := \Big\{ w \in \mathbb{R}^n : \exists\, (t_k, r_k) \downarrow (0,0),\; w_k \to w,\; t_k/r_k \to 0,\; \bar{x} + t_k d + \tfrac{1}{2} t_k^2 r_k w_k \in S \forall k \Big\}$$

This construction is always a cone containing $0$, even if the Bouligand outer set is empty or fails to be a cone (Ouyang et al., 26 Apr 2024, Ma et al., 16 Jul 2025).

2. Fundamental Properties and Theoretical Landscape

Set-theoretic and Topological Features

• $T^2_S(\bar{x}; d)$ is closed but can lack conicity or convexity and may be empty even if $S$ is convex.
• $\widetilde{T}^2_S(\bar{x}; d)$ is a closed cone, nonempty for every $d \in T_S(\bar{x})$; moreover, $$T^2_S(\bar{x}; d) \subset \widetilde{T}^2_S(\bar{x}; d)$$ (Ma et al., 16 Jul 2025, Ouyang et al., 26 Apr 2024).
• Homogeneity: For any scalar $\tau > 0$, $T^2_S(\bar{x}; \tau d) = T^2_S(\bar{x}; d)$, and similarly for $\widetilde{T}^2_S$.
• Penot’s key result indicates that at least one of $T^2_S(\bar{x}; d)$ or $\widetilde{T}^2_S(\bar{x}; d) \setminus \{0\}$ is nonempty for any $d \in T_S(\bar{x})$.

Classical and Nonstandard Analysis Perspective

• Classical approach uses limits of secant sets (Painlevé–Kuratowski) or epi-derivatives, relying on sequences approaching the reference point (Chen et al., 2019, Durea et al., 2011).
• The nonstandard formulation by Kutateladze (Kutateladze, 2020) employs infinitesimal neighborhoods in Internal Set Theory, yielding a characterization via inclusion of triples of shifted points:
  • For $F \subset X$, $x \in F$, and directions $v_1, v_2$, $v$ is in the second-order Clarke-type tangent set $Cl^{(2)}(F, x)(v_1, v_2)$ exactly if for all standard $\epsilon_1, \epsilon_2 > 0$ and $x' \approx x$,

  $$x' + \epsilon_1 v_1 \in F,\quad x' + \epsilon_2 v_2 \in F,\quad x' + \epsilon_1 v_1 + \epsilon_2 v_2 + 4\epsilon_1 \epsilon_2 v \in F$$

This approach provides a unified and quantifier-minimal test of higher-order tangency.

3. Calculus of Second-Order Tangent Sets and Metric Subregularity

Recent advances exploit metric subregularity—a local error-bound condition—to derive inclusion-type calculus rules for second-order tangent sets, circumventing the compactness requirements typical in classical variance analysis (Durea et al., 2011). Key developments:

• Inverse Image Calculus: If $D \subset X$, $E \subset Y$, and $f: X \to Y$ is $C^2$, then under metric subregularity,

$$T^2_B(D \cap f^{-1}(E), \bar{x}, x_1) \supset T^2_B(D, \bar{x}, x_1) \cap (f'(\bar{x}))^{-1} \left[ T^2_B(E, f(\bar{x}), f'(\bar{x}) x_1) - \tfrac{1}{2} f''(\bar{x})(x_1, x_1) \right]$$

• Sum Rule for Derivatives: For two set-valued maps and a composite structure, the second-order Bouligand derivative admits an inclusion formula, facilitating analysis of perturbation maps and constraint structures without compactness (Durea et al., 2011).
• Perturbation Maps: For structural constraints of the form $z \in F(x, y) + K(x, y)$, the second-order tangent set to the graph of the induced map is inclusively described by the summation of Bouligand tangent sets to each component (Durea et al., 2011).

This calculus is essential for both the existence and computation of directional second-order derivatives in optimization and control.

4. Second-Order Tangent Sets in Surface Flexes and Nonrigidity

In differential geometry, second-order tangent sets provide the analytic setting for the paper of flexes and nonrigidity of surfaces in $\mathbb{R}^3$ (Alexandrov, 2021). The construction proceeds via isometric deformations:

• The space of smooth surfaces $X$ is the set of all smooth boundary-free 2-dimensional immersions.
• The subset of nonrigid surfaces $N \subset X$ consists of those admitting a nontrivial first-order flex.
• The first-order tangent set at $x$ is

$$T_N(x) := \{ \xi \in \Gamma(x^* T \mathbb{R}^3) : D F(x)[\xi] = 0 \}$$

where $F(x)$ is the first fundamental form and $D F$ its linearization.

• The second-order tangent set $T_N^2(x)$ is comprised of pairs $(\xi, w)$ satisfying

$D^2 F(x)[\xi, \xi] + D F(x)[w] = 0$

These equations encode the extension of infinitesimal flexes to second-order isometric deformations, crucial for analyzing the local geometry of the set of nonrigid surfaces.

• The extension theorem proves that any first-order flex tangent to $N$ extends to a second-order flex, under smoothness assumptions (Alexandrov, 2021).

5. Exact Formulas and Applications in SOC Complementarity Problems

For sets governed by complementarity or cone constraints (notably the SOC cone complementarity set), second-order tangent sets are vital for variational analysis and optimality conditions (Chen et al., 2019). Salient points:

• The SOC complementarity set

$$\mathcal{Q} = \{ (x, y) \in K \times K : \langle x, y \rangle = 0 \}$$

is nonconvex and not a union of finitely many polyhedral convex sets.

• The exact formula for the second-order tangent set $T^2_{\mathcal{Q}}((x, y); (d, w))$ is derived for six prototypical cases, leveraging the directional derivatives of the projector operator over the cone.
• For each reference case (interior, boundary, origin), the associated quadratic and linear constraints defining the second-order tangent set are explicitly spelled out.
• Application: Second-order necessary conditions for mathematical programs with SOC complementarity constraints (MPCC-SOC) utilize the calculated second-order tangent sets in the curvature term of the Lagrangian, yielding sharp necessary conditions without reliance on convexity or polyhedrality.

6. Role in Second-Order Optimality and Nonconvex Constraints

Second-order tangent sets—especially the outer tangent and asymptotic cones—are foundational for expressing necessary and sufficient second-order optimality conditions in nonconvex set-constrained problems (Ouyang et al., 26 Apr 2024, Ma et al., 16 Jul 2025). Highlights include:

• Support Functions: Second-order curvature terms in Lagrange necessary conditions are written as the support function $\sigma_{T^2}$ (or lower generalized $\hat{\sigma}$), connecting the set-valued geometry directly to dual variables.
• Weak Sharp Minima: The framework accommodates conditions for weak sharp minima and “directional optimality” even when the outer second-order tangent set is nonconvex or empty, greatly relaxing the assumptions required in previous convex-analytic approaches.
• Sufficiency Without Regularity: Sufficient second-order optimality can be proven without uniform second-order regularity or critical cone approximation, by appealing to the properties of the asymptotic cone and the lower generalized support.

7. Nonstandard Analysis Approach and Unification

Kutateladze’s construction via Internal Set Theory (Kutateladze, 2020) yields an alternative, infinitesimal-based concept of second-order tangent sets. Noteworthy features:

• Nonstandard formulation replaces classical $\epsilon$–$\delta$–$t \to 0$ arguments with quantifier-reduced infinitesimal ones, rendering complex limit processes tractable.
• The second-order Clarke-type tangent set is characterized by a single uniform inclusion (“three points belong to $F$ for all standard positive $\alpha, \beta$”), providing a unification of classical and Clarke-type parabolic cones.
• The framework recovers classical results for convex cones and smooth manifolds and establishes properties such as closedness and semigroup structure for the second-order set.

Table: Comparison of Outer vs. Asymptotic Second-Order Tangent Sets

Property	Outer Second-Order Tangent Set $T^2_S$	Asymptotic Cone $\widetilde{T}^2_S$
Closedness	Always closed	Always closed
Conicity	Not necessarily a cone	Always a cone
Nonconvexity/Emptiness Possible	Yes	No (always contains 0)
Inclusion	$T^2_S \subset \widetilde{T}^2_S$	N/A
Use in Optimality Conditions	Curvature terms via support functions	Provides backup directions if $T^2_S$ empty

Summary

Second-order tangent sets are central to modern variational analysis and set-valued calculus, underpinning the paper of second-order phenomena in geometry and optimization. They possess intricate set-theoretic, topological, and analytic properties, with multiple formulations suited to both regular and nonregular, convex and nonconvex, finite- and infinite-dimensional settings. The development of exact formulas, calculus rules under metric subregularity, and nonstandard characterizations has substantially broadened their applicability. Their role in expressing second-order necessary and sufficient conditions for optimality, especially when classical regularity fails, is foundational for the advancement of nonlinear analysis.