We are interested in the following question (definitions and references are given below):

Main Question: Given an upper-semicontinuous polyhedral multifunction $F:R^n \rightarrow R^m$, is there always a Lipschitz continuous function $g:R^n \rightarrow R^m$ such that $g(x) \in F(x)$ for all $x \in R^n$ ?

In general, *upper* semicontinuity is probably not the right property to guarantee continuous selections (see [HP-1], page 89). On the other hand, polyhedrality of the multifunction may help.

Related Question: Under what *mild* conditions is a positive answer to the above question guaranteed?

Motivation: Solutions of finite-dimensional variational inequalities (VI) or complementarity problems (CP) (see [FP-1]) typically have upper semicontinuous solution maps. Furthermore, if the functions and sets defining the VI or CP have affine or linear (or polyhedral) structure, their solution maps are polyhedral multifunctions. The main question posed above is then a natural question to ask in the study of differential inclusions $\dot{x} \in G(x)$ that have these solution maps appearing in the definition of $G(x)$.

Definitions:

- A
*multifunction*is simply a*set-valued*map. - A multifunction $F:R^n \rightarrow R^m$ is said to be
*upper semicontinuous*at a point $\bar{x}$ if for every open set $\mathcal{V}$ containing $F(\bar{x})$, there exists an open neighbourhood $\mathcal{U}$ of $\bar{x}$ such that, for each $x \in \mathcal{U}$, $\mathcal{V}$ contains $F(x)$. - A multifunction is said to be
*polyhedral*if its graph is a polyhedral subset of $R^{n + m}$.

Referenecs:

- [FP-1] F. Facchinei and J-S Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems (vol. 1), pp. 138-139.
- [HP-1] S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis (vol. 1), p. 36 and p. 89.
- [M-1] Ernest Michael, Continuous Selections I, The Annals of Mathematics, Vol. 63, (1956), pp. 361-382.
- [M-2] Ernest Michael, Continuous Selections II, The Annals of Mathematics, Vol. 64, (1956), pp. 562-580.
- [M-2] Ernest Michael, Continuous Selections III, The Annals of Mathematics, Vol. 65, (1957), pp. 375-390.