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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.

Let $s=Kx$ and assume $s>0$. The scalar Poisson data fidelity integrand has derivative $1-y/s$. Applying the permitted interchange of differentiation and integration in a direction $z$ givesThus the ambient-space Frechet derivative, or its Hilbert space gradient when the pairing is an inner product, isThe quotient is pointwise, and the adjoint operator transports it back from the data domain to the source domain. The chosen spaces must ensure the integrals and derivative pairing are finite; an output bounded away from zero is a useful sufficient condition.

The printed set of probability density functions and positive cone are not vector spaces, so the notation for a bounded linear map between them is understood as a positive bounded linear operator on ambient function spaces, restricted to admissible inputs. If unit mass is a constraint, admissible directions satisfy ${\int }_{\Omega }z=0$. The displayed gradient is then a representative modulo an additive constant; optimization with that constraint additionally includes its normal cone.