Swarm dynamics for global optimization 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 $P\left(V\right)$ of probability distributions on $V$ and we penalize it with a time-dependent generalized entropy functional. Endowing $P\left(V\right)$ 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 $P\left(V\right)$-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 et al. (2023), but here we go further by considering more general generalized 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.