Variational analysis

Variational analysis studies optimization, perturbation stability and generalized differentiation through functions, sets and set-valued mappings. Convex analysis is a foundational special case; limiting normal constructions also handle nonconvex geometry.

Table of contents
- Normal cone Variational analysis
  - Limiting normal cone Normal cone
  - Fréchet normal cone Normal cone
- Set-valued analysis Variational analysis
  - Set-valued mapping Set-valued analysis
    - Aubin property Set-valued mapping
      - Mordukhovich criterion Aubin property
    - Graph of a set-valued mapping Set-valued mapping
      - Limiting coderivative Graph of a set-valued mapping

Normal cone ( $N_{C} (z)$ )

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Normal cones describe generalized supporting directions to sets. For a convex set, the normal condition is

⟨ w, z^{'} - z ⟩ \leq 0

for every

z^{'}

in the set. For nonconvex sets, the Fréchet normal cone and the limiting normal cone distinguish local regular support from limits of such supports.

Limiting normal cone ( $N_{C} (z) = lim sup_{z^{'} \to z} N_{C} (z^{'})$ )

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The limiting normal cone consists of limits

w_{k} \to w

of vectors from the Fréchet normal cone

w_{k} \in N_{C} (z_{k})

with

z_{k} \to z

C

. It need not be convex. Including normals from nearby graph pieces is essential to the Mordukhovich criterion.

Fréchet normal cone ( $N_{C} (z)$ )

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A vector

w

is a regular normal at

z \in C

when

lim sup_{z^{'} \to z, z^{'} \in C ∖ {z}} ⟨ w, z^{'} - z ⟩ /∥ z^{'} - z ∥ \leq 0

. At an isolated point all vectors satisfy this condition. It is the local first-order supporting cone.

Set-valued analysis studies relations whose output at one input is a set. Solution maps, subdifferentials and generalized normal mappings are important examples. Graph geometry provides definitions of continuity, local stability and differentiation.

Set-valued mapping ( $S : X ⇉ Y$ )

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A mapping

S : X ⇉ Y

assigns a subset

S (u) \subset Y

to each

u

. Empty and multiple values are allowed. The graph of a set-valued mapping records the relation as a subset of

X \times Y

Aubin property ( $S (u) \cap V \subset S (u^{'}) + κ ∥ u - u^{'} ∥ B$ )

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Near

(\overset{u}{ˉ}, \overset{v}{ˉ})

in a graph, this property requires neighborhoods

U, V

and a finite

κ

such that

S (u) \cap V \subset S (u^{'}) + κ ∥ u - u^{'} ∥ B

for all

u, u^{'} \in U

. Nearby solutions can be matched under nearby perturbations with linear displacement control. Single-valued maps reduce to local Lipschitz continuity.

Mordukhovich criterion ( $D^{*} S (\overset{u}{ˉ} ∣ \overset{v}{ˉ}) (0) = {0}$ )

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In finite dimensions, if the graph is locally closed at

(\overset{u}{ˉ}, \overset{v}{ˉ})

, the Aubin property holds exactly when

D^{*} S (\overset{u}{ˉ} ∣ \overset{v}{ˉ}) (0) = {0}

. Thus no nonzero horizontal vector in the limiting normal cone may occur. The exact local Lipschitz modulus is the outer norm of this coderivative.

Graph of a set-valued mapping ( $gph S = {(u, v) : v \in S (u)}$ )

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The graph is

gph S = {(u, v) : v \in S (u)}

. Local graph normals encode the Mordukhovich coderivative and hence the Aubin property. Graph closedness is a local hypothesis in the sensitivity criterion.

Limiting coderivative ( $D^{*} S (\overset{u}{ˉ} ∣ \overset{v}{ˉ})$ )

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The limiting coderivative is

D^{*} S (\overset{u}{ˉ} ∣ \overset{v}{ˉ}) (v^{*}) = {u^{*} : (u^{*}, - v^{*}) \in N_{gph S} (\overset{u}{ˉ}, \overset{v}{ˉ})}

. The negative sign on the output dual is essential. For a smooth single-valued function it gives the transpose Jacobian acting on

v^{*}

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Variational analysis is a branch of mathematics that deals with the study of optimization and equilibrium problems, particularly in the context of functional analysis and differential inclusions. It provides a framework for analyzing problems where one seeks to minimize or maximize objective functions, often subject to certain constraints.

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