A policy iteration algorithm for fixed point problems with nonexpansive operators

Abstract

The aim of this paper is to solve the fixed point problems:

$$ v = \mathcal{O}v,\quad \hbox{with}\, \mathcal{O}v(x) \mathop{=}^{\rm def} \max (Lv(x), Bv(x) ), x \in \varepsilon, \quad (1)$$

where $\varepsilon$ is a finite set, L is contractive and B is a nonexpansive operator and

$$ v = \mathcal{O}v,\quad \hbox{with} \mathcal{O}v(x) \mathop{=}^{\rm def} \max\left(\sup_{w \in \mathcal{W}} L^{w} v(x) ,\sup_{z \in \mathcal{Z}} B^{z} v(x)\right), x \in \varepsilon, \quad (2)$$

where $\mathcal{W}$ and $\mathcal{Z}$ are general control sets, the operators L ^w are contractive and operators B ^z are nonexpansive. For these two problems, we give conditions which imply existence and uniqueness of a solution and provide a policy iteration algorithm which converges to the solution. The proofs are slightly different for the two problems since the set of controls is finite for (1) while it is not necessary the case for problem (2). Equation (2) typically arises in numerical analysis of quasi variational inequalities and variational inequalities associated to impulse or singular stochastic control.