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A policy iteration algorithm for fixed point problems with nonexpansive operators

Author

Listed:
  • Jean-Philippe Chancelier
  • Marouen Messaoud
  • Agnès Sulem

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. Copyright Springer-Verlag 2007

Suggested Citation

  • Jean-Philippe Chancelier & Marouen Messaoud & Agnès Sulem, 2007. "A policy iteration algorithm for fixed point problems with nonexpansive operators," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(2), pages 239-259, April.
    • Handle: RePEc:spr:mathme:v:65:y:2007:i:2:p:239-259
    DOI: 10.1007/s00186-006-0103-3
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    Citations

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    Cited by:

    1. Parsiad Azimzadeh & Peter A. Forsyth, 2015. "Weakly chained matrices, policy iteration, and impulse control," Papers 1510.03928, arXiv.org, revised Sep 2017.
    2. Xu, Ran & Woo, Jae-Kyung, 2020. "Optimal dividend and capital injection strategy with a penalty payment at ruin: Restricted dividend payments," Insurance: Mathematics and Economics, Elsevier, vol. 92(C), pages 1-16.
    3. Diego Zabaljauregui, 2020. "Optimal market making under partial information and numerical methods for impulse control games with applications," Papers 2009.06521, arXiv.org.
    4. Johannes Muhle-Karbe & Max Reppen & H. Mete Soner, 2016. "A Primer on Portfolio Choice with Small Transaction Costs," Papers 1612.01302, arXiv.org, revised May 2017.
    5. Diego Zabaljauregui, 2019. "A fixed-point policy-iteration-type algorithm for symmetric nonzero-sum stochastic impulse control games," Papers 1909.03574, arXiv.org, revised Jun 2020.

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