Second-Order Limiting Subdifferential

• Second-order limiting subdifferential is a generalized Hessian for nonsmooth functions, providing a robust framework for second-order variational analysis.
• It employs coderivatives and normal cones to establish comprehensive calculus rules and rigorous optimality conditions in nonconvex settings.
• This construction underpins advanced topics such as tilt stability, critical multiplier analysis, and strong metric regularity in optimization.

A second-order limiting subdifferential is the central generalized second-order object in nonsmooth variational analysis, serving as the canonical extension of the classical Hessian for lower semicontinuous, nonconvex, and nonsmooth functions. Originating with the works of Mordukhovich and Rockafellar, it underpins much of contemporary nonsmooth optimization, variational stability, and generalized differentiation theory. This construction enables precise second-order optimality conditions, comprehensive calculus rules, and characterizations of criticality, stability, and metric regularity in a broad class of nonsmooth problems (Khanh et al., 2023, Mordukhovich et al., 2011, Drusvyatskiy et al., 2013).

1. Definition and Construction

Let $f:\mathbb{R}^n \to \mathbb{R}\cup\{+\infty\}$ be a proper lower semicontinuous function, prox-regular at $\bar x$ for $\bar v \in \partial f(\bar x)$. The first-order (Mordukhovich/limiting) subdifferential is

$$\partial f(x) := \limsup_{(x',v')\to(x,v),\, v'\in\hat\partial f(x')} \{ v'\},$$

where $\hat\partial f$ denotes the Fréchet subdifferential. The graph of $\partial f$ is

$$\mathrm{gph}\,\partial f = \{\, (x,v): v\in\partial f(x)\, \}.$$

The second-order limiting subdifferential (generalized Hessian) of $f$ at $(\bar x,\bar v)$ in direction $w\in\mathbb{R}^n$ is defined (Khanh et al., 2023, Mordukhovich et al., 2011, Drusvyatskiy et al., 2013) as

$\bar x$0

where $\bar x$1 denotes the Mordukhovich coderivative and $\bar x$2 is the limiting normal cone to the graph.

In the $\bar x$3-smooth case, $\bar x$4, recovering the classical Hessian (Khanh et al., 2023, Drusvyatskiy et al., 2013).

2. Key Properties

The second-order limiting subdifferential possesses several structural and analytical properties (Khanh et al., 2023, Mordukhovich et al., 2011, Tuyen, 2 Mar 2025, Huy et al., 2019):

• Closed Graph: $\bar x$5 is closed and conic in $\bar x$6.
• Homogeneity: $\bar x$7 implies $\bar x$8.
• Local Boundedness: The mapping $\bar x$9 is locally bounded and upper semicontinuous.
• Positive (Semi)definiteness: At a local minimizer $\bar v \in \partial f(\bar x)$0 with $\bar v \in \partial f(\bar x)$1, every $\bar v \in \partial f(\bar x)$2 satisfies $\bar v \in \partial f(\bar x)$3 for all $\bar v \in \partial f(\bar x)$4; strict definiteness characterizes tilt stability.
• Reduction to Hessian: When $\bar v \in \partial f(\bar x)$5 is $\bar v \in \partial f(\bar x)$6 at $\bar v \in \partial f(\bar x)$7, $\bar v \in \partial f(\bar x)$8.
• Nonemptiness and Compactness: For $\bar v \in \partial f(\bar x)$9 functions, $$\partial f(x) := \limsup_{(x',v')\to(x,v),\, v'\in\hat\partial f(x')} \{ v'\},$$0 is a nonempty compact subset of $$\partial f(x) := \limsup_{(x',v')\to(x,v),\, v'\in\hat\partial f(x')} \{ v'\},$$1 (Tuyen, 2 Mar 2025).

3. Calculus Rules

Second-order limiting subdifferentials allow robust calculus, supporting sum, chain, and scalar multiplication rules under suitable regularity conditions (Khanh et al., 2023, Mordukhovich et al., 2011, Huy et al., 2019, Sadygov, 2017):

Operation	Formula (under qualification conditions)	Context
Sum	$$\partial f(x) := \limsup_{(x',v')\to(x,v),\, v'\in\hat\partial f(x')} \{ v'\},$$2	$$\partial f(x) := \limsup_{(x',v')\to(x,v),\, v'\in\hat\partial f(x')} \{ v'\},$$3 prox-regular at $$\partial f(x) := \limsup_{(x',v')\to(x,v),\, v'\in\hat\partial f(x')} \{ v'\},$$4, $$\partial f(x) := \limsup_{(x',v')\to(x,v),\, v'\in\hat\partial f(x')} \{ v'\},$$5
Chain	$$\partial f(x) := \limsup_{(x',v')\to(x,v),\, v'\in\hat\partial f(x')} \{ v'\},$$6	$$\partial f(x) := \limsup_{(x',v')\to(x,v),\, v'\in\hat\partial f(x')} \{ v'\},$$7 $$\partial f(x) := \limsup_{(x',v')\to(x,v),\, v'\in\hat\partial f(x')} \{ v'\},$$8, $$\partial f(x) := \limsup_{(x',v')\to(x,v),\, v'\in\hat\partial f(x')} \{ v'\},$$9 prox-regular
Scalar Mult.	$\hat\partial f$0	$\hat\partial f$1

Equality in the sum and chain rules requires stronger conditions (e.g., full rank, graphical regularity) (Mordukhovich et al., 2011, Drusvyatskiy et al., 2013).

4. Second-Order Optimality and Stability

The second-order limiting subdifferential characterizes both necessary and sufficient optimality conditions in nonsmooth and possibly nonconvex problems (Khanh et al., 2023, Drusvyatskiy et al., 2013, Drusvyatskiy et al., 2012):

• Necessary Condition: If $\hat\partial f$2 is a local minimizer and $\hat\partial f$3,

$\hat\partial f$4

• Sufficient Condition: If $\hat\partial f$5 s.t.\ $\hat\partial f$6 for all nearby $\hat\partial f$7, $\hat\partial f$8, then $\hat\partial f$9 is a strict local minimizer.
• Tilt Stability: If $\partial f$0 is subdifferentially continuous and there is $\partial f$1 so that

$\partial f$2

then $\partial f$3 is tilt-stable with modulus $\partial f$4, i.e., the argmin mapping $\partial f$5 is single-valued and Lipschitz (Khanh et al., 2023, Drusvyatskiy et al., 2012, Drusvyatskiy et al., 2013).

5. Explicit Computations and Classical Cases

For convex piecewise-linear (PL) functions and $\partial f$6 functions, the second-order subdifferential admits explicit expressions (Khanh et al., 2023, Mordukhovich et al., 2011, Mordukhovich et al., 2016):

• Convex PL: $\partial f$7 yields $\partial f$8 and

$\partial f$9

for all $$\mathrm{gph}\,\partial f = \{\, (x,v): v\in\partial f(x)\, \}.$$0, reflecting the absence of curvature.

• $$\mathrm{gph}\,\partial f = \{\, (x,v): v\in\partial f(x)\, \}.$$1 Reduction: For $$\mathrm{gph}\,\partial f = \{\, (x,v): v\in\partial f(x)\, \}.$$2 $$\mathrm{gph}\,\partial f = \{\, (x,v): v\in\partial f(x)\, \}.$$3 at $$\mathrm{gph}\,\partial f = \{\, (x,v): v\in\partial f(x)\, \}.$$4,

$$\mathrm{gph}\,\partial f = \{\, (x,v): v\in\partial f(x)\, \}.$$5

• PLQ Functions and Compositions: For fully amenable composite functions $$\mathrm{gph}\,\partial f = \{\, (x,v): v\in\partial f(x)\, \}.$$6 with $$\mathrm{gph}\,\partial f = \{\, (x,v): v\in\partial f(x)\, \}.$$7 $$\mathrm{gph}\,\partial f = \{\, (x,v): v\in\partial f(x)\, \}.$$8, $$\mathrm{gph}\,\partial f = \{\, (x,v): v\in\partial f(x)\, \}.$$9 convex PLQ, the chain rule applies and computable formulas for $f$0 and critical cones are available (Mordukhovich et al., 2011, Mordukhovich et al., 2016).

6. Applications: Constrained Optimization, Critical Multipliers, and Stability

The second-order limiting subdifferential underpins advanced topics in optimization, including KKT multipliers, error bounds, stability, and variational inclusions (Tuyen, 2 Mar 2025, Mordukhovich et al., 2016, Mordukhovich et al., 2011):

• KKT Systems: Second-order necessary and sufficient KKT-type conditions for $f$1 vector optimization with constraints require the presence of $f$2 in multiplier and critical cone expressions (Tuyen, 2 Mar 2025).
• Critical Multipliers: The criticality of Lagrange multipliers in variational/KKT systems is characterized via the second-order limiting subdifferential, with explicit conic-algebraic criteria for piecewise-linear constraints (Mordukhovich et al., 2016).
• Strong Regularity, Quadratic Growth, and Tilt Stability: Weak and strong metric regularity, uniform quadratic growth, and tilt stability are all characterized via positive definiteness of the second-order (limiting) subdifferential (Drusvyatskiy et al., 2012, Drusvyatskiy et al., 2013). The equivalences enable robust stability and error-bound guarantees in nonconvex and nonsmooth settings.

7. Relations to Other Second-Order Constructions

The second-order limiting subdifferential is closely related to, but generally finer than, other second-order objects:

• Clarke’s Generalized Hessian: In $f$3 settings, the Mordukhovich construction refines Clarke’s generalized Jacobian by capturing additional directional information ("hidden curvature") provided by the normal cone to the graph (Huy et al., 2019, Tuyen, 2 Mar 2025).
• Second Subderivative: The variational second subderivative $f$4 relates closely to the coderivative-based $f$5 and provides variational characterizations of tilt stability and quadratic growth (Drusvyatskiy et al., 2012, Drusvyatskiy et al., 2013).
• Operator Inclusions and Abstract Nonsmooth Calculus: The second-order limiting subdifferential extends to Banach spaces and set-valued mappings, supporting extremal conditions in operator-inclusion problems (Sadygov, 2017).

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References:

• "Second-Order Subdifferential Optimality Conditions in Nonsmooth Optimization" (Khanh et al., 2023)
• "Second-order subdifferential calculus with applications to tilt stability in optimization" (Mordukhovich et al., 2011)
• "Second-order growth, tilt stability, and metric regularity of the subdifferential" (Drusvyatskiy et al., 2013)
• "On second-order Karush--Kuhn--Tucker optimality conditions for $f$6 vector optimization problems" (Tuyen, 2 Mar 2025)
• "Second-order optimality conditions for multiobjective optimization problems with constraints" (Huy et al., 2019)
• "Tilt stability, uniform quadratic growth, and strong metric regularity of the subdifferential" (Drusvyatskiy et al., 2012)
• "Critical Multipliers in Variational Systems via Second-order Generalized Differentiation" (Mordukhovich et al., 2016)
• "Second-order subdifferential. Extremal problems for operational inclusions" (Sadygov, 2017)