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For $v\in Z$, write $M(v)={\max }_i{v}_i$. Maximizing a linear functional on the price simplex givesThus $g(v)$ is a nonempty closed convex face of the compact price simplex. Its graph of a set-valued mapping is closed: if $v_n\to v$, $p_n\to p$ and $p_n\in g(v_n)$, then, for every fixed $q\in P$, the inequalities $p_n\cdot v_n\ge q\cdot v_n$ pass to $p\cdot v\ge q\cdot v$.

Each value $z(p)$ is closed in $Z$ by the closed-graph hypothesis, hence compact; it is nonempty and convex by assumption. Therefore $F(p,v)=g(v)\times z(p)$ has nonempty closed convex values in the nonempty compact convex set $K=P\times Z$.

Its graph of a set-valued mapping is also closed. If $(p_n,v_n)\to (p,v)$ and $(q_n,w_n)\to (q,w)$ with $q_n\in g(v_n)$ and $w_n\in z(p_n)$, the two closed-graph properties give $q\in g(v)$ and $w\in z(p)$. Finally, this closed graph implies upper hemicontinuity because the whole output space $K$ is compact. If an open set containing $F(p,v)$ failed to contain the nearby values, select outputs outside it along a convergent input sequence. A subsequence of those outputs converges in $K$; closedness puts its limit in $F(p,v)$, whereas closedness of the open set's complement puts it outside. This contradiction proves upper hemicontinuity.

Consequently all the hypotheses of the Kakutani fixed-point theorem hold: the domain is nonempty, compact and convex, and the set-valued mapping maps it into itself with nonempty closed convex values and upper hemicontinuity.

Work in finite-dimensional Euclidean spaces. Suppose the graph of a set-valued mapping,is locally closed at $(\bar{u},\bar{v})\in \mathrm{gph}S$. Local closedness means that its intersection with some neighborhood of this point is closed relative to that neighborhood. This assumption is part of the criterion.

For a set $C$ and $z\in C$, define the Fréchet normal cone byAt an isolated point the condition is vacuous, so all vectors are regular normals. The limiting normal cone consists of limits of these regular normals at nearby points:It can be nonconvex. The Mordukhovich coderivative isThe minus sign in the output-dual component is part of the definition. For a single-valued continuously differentiable mapping $s$, the graph normal is $(Ds(\bar{u}{)}^T{v}^{\ast },-v^{\ast })$, giving $D^{\ast }s(\bar{u}\mid s(\bar{u}))(v^{\ast })=\{Ds(\bar{u}{)}^T{v}^{\ast }\}$. Thus the coderivative extends a transpose derivative, rather than the ordinary forward derivative.

The exact Mordukhovich criterion isThis is equivalent to excluding a nonzero horizontal limiting graph normal $(u^{\ast },0)$. One must use the limiting normal cone; simply checking regular normals at the reference point can miss normals inherited from neighboring graph pieces.

For sensitivity analysis, apply this test to a solution map or use coderivative and normal cone calculus to express its graph normals through optimality constraints. Triviality of this kernel then proves Lipschitz-like stability without solving the perturbed problem explicitly. Under these same finite-dimensional, locally closed assumptions, the exact Lipschitz modulus is the outer normThe zero-kernel condition is the qualitative criterion; the modulus quantifies the sensitivity bound. For the next problem an explicit local formula is simpler than evaluating these graph normals.

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 $F$ is upper hemicontinuous at $x$ when every open set containing $F(x)$ also contains $F(x^{\prime })$ for all nearby $x^{\prime }$. With values in a fixed compact space, a closed graph of a set-valued mapping implies upper hemicontinuity: a violating sequence has a convergent subsequence of outputs whose limit lies in the limiting value.