Assigning Stationary Distributions to Sparse Stochastic Matrices

Abstract

The target stationary distribution problem (TSDP) is the following: given an irreducible stochastic matrix $G$ and a target stationary distribution $\hat μ$, construct a minimum norm perturbation, $Δ$, such that $\hat G = G+Δ$ is also stochastic and has the prescribed target stationary distribution, $\hat μ$. In this paper, we revisit the TSDP under a constraint on the support of $Δ$, that is, on the set of non-zero entries of $Δ$. This is particularly meaningful in practice since one cannot typically modify all entries of $G$. We first show how to construct a feasible solution $\hat G$ that has essentially the same support as the matrix $G$. Then we show how to compute globally optimal and sparse solutions using the component-wise $\ell_1$ norm and linear optimization. We propose an efficient implementation that relies on a column-generation approach which allows us to solve sparse problems of size up to $10^5 \times 10^5$ in a few minutes. We illustrate the proposed algorithms with several numerical experiments.