Set-valued analysis

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.

A mapping $S:X\rightrightarrows Y$ 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 $X\times Y$.

A set-valued mapping $F:X\rightrightarrows Y$ is upper hemicontinuous at $x$ when every open set $U$ containing $F(x)$ also contains $F(x^{\prime })$ for all $x^{\prime }$ in some neighborhood of $x$. Empty values are permitted. For a single-valued map this is the usual neighborhood definition of continuity.

If $Y$ is compact and the graph of a set-valued mapping is closed, upper hemicontinuity follows without a sequential-countability assumption. For each $y\in Y\setminus U$, the point $(x,y)$ is outside the closed graph, so there is a product neighborhood $N_y\times V_y$ disjoint from it. Finitely many $V_y$ cover the compact set $Y\setminus U$. Intersect their $N_y$: every nearby value is then contained in $U$.

Compact control is essential for this implication. The graph of $f(x)=1/x$ for $x\ne 0$, $f(0)=0$, is closed in ${\mathbb{R}}^2$, but the corresponding singleton-valued map is not upper hemicontinuous at zero. Its nearby outputs escape every bounded neighborhood of zero.

Near $(\bar{u},\bar{v})$ in a graph, this property requires neighborhoods $U,V$ and a finite $\kappa$ such that $S(u)\cap V\subset S(u^{\prime })+\kappa \Vert u-u^{\prime }\Vert \mathbb{B}$ for all $u,u^{\prime }\in U$. Nearby solutions can be matched under nearby perturbations with linear displacement control. Single-valued maps reduce to local Lipschitz continuity.

In finite dimensions, if the graph is locally closed at $(\bar{u},\bar{v})$, the Aubin property holds exactly when $D^{\ast }S(\bar{u}\mid \bar{v})(0)=\{0\}$. Thus no nonzero horizontal vector in the limiting normal cone may occur. The exact local Lipschitz modulus is the outer norm of this coderivative.

The graph is $\mathrm{gph}S=\{(u,v):v\in S(u)\}$. Local graph normals encode the Mordukhovich coderivative and hence the Aubin property. Graph closedness is a local hypothesis in the sensitivity criterion.

The limiting coderivative is $D^{\ast }S(\bar{u}\mid \bar{v})(v^{\ast })=\{u^{\ast }:(u^{\ast },-v^{\ast })\in N_{\mathrm{gph}S}(\bar{u},\bar{v})\}$. The negative sign on the output dual is essential. For a smooth single-valued function it gives the transpose Jacobian acting on $v^{\ast }$.