On the free boundary of a quasivariational inequality arising in a problem of quality control

Abstract:

In some recent work in stochastic optimization with partial observation occurring in quality control problems, Anderson and Friedman [1], [2] have shown that the optimal cost can be determined as a solution of the quasi variational inequality \[ \begin {gathered} Mw\left ( p \right ) + f\left ( p \right ) \geq 0, w\left ( p \right ) \leq \psi \left ( {p; w} \right ), \left ( {Mw\left ( p \right ) + f\left ( p \right )} \right )\left ( {w\left ( p \right ) - \psi \left ( {p; w} \right )} \right ) = 0 \end {gathered} \] in the simplex ${p_i} > 0$, $\sum \nolimits _{i = 1}^n {{p_i} = 1}$. Here f, $\psi$ are given functions of p, $\psi$ is a functional of w, and M is a given elliptic operator degenerating on the boundary. This system has a unique solution when M does not degenerate in the interior of the simplex. The aim of this paper is to study the free boundary, that is, the boundary of the set where $w\left ( p \right ) < \psi \left ( {p; w} \right )$.