Nhat-Thang Le | Searcharxiv

arXiv2026

A forward algorithm for a class of Markov zero-sum stopping games

In this paper, we propose a new efficient algorithm to compute the value function for zero-sum stopping games featuring two players with opposing interests. This can be seen as a game version of the ''forward algorithm'' for (one-player) optimal stopping problem, first introduced by Irle [6] for discrete-time Markov processes and later revisited by Miclo \& Villeneuve [8] for continuous-time Markov processes on general state spaces. This paper focuses on a game driven by a homogeneous Markov process taking values in a finite state space and also discusses about the number of iterations needed. Illustrated computational implementations for a few particular examples are also provided.

math.PR↗

arXiv2024

Swarm dynamics for global optimisation on finite sets

Consider the global optimisation of a function $U$ defined on a finite set $V$ endowed with an irreducible and reversible Markov generator.By integration, we extend $U$ to the set $\mathcal{P}(V)$ of probability distributions on $V$ and we penalise it with a time-dependent generalised entropy functional.Endowing $\mathcal{P}(V)$ with a Maas' Wasserstein-type Riemannian structure, enables us to consider an associated time-inhomogeneous gradient descent algorithm.There are several ways to interpret this $\cP(V)$-valued dynamical system as the time-marginal laws of a time-inhomogeneous non-linear Markov process taking values in $V$, each of them allowing for interacting particle approximations.This procedure extends to the discrete framework the continuous state space swarm algorithm approach of Bolte, Miclo and Villeneuve \cite{Bolte}, but here we go further by considering more general generalised entropy functionals for which functional inequalities can be proven.Thus in the full generality of the above finite framework, we give conditions on the underlying time dependence ensuring the convergence of the algorithm toward laws supported by the set of global minima of $U$.Numerical simulations illustrate that one has to be careful about the choice of the time-inhomogeneous non-linear Markov process interpretation.

math.FA↗