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The target stationary distribution problem (TSDP) is the following: given anirreducible stochastic matrix $G$ and a target stationary distribution $\hat\mu$, construct a minimum norm perturbation, $\Delta$, such that $\hat G =G+\Delta$ is also stochastic and has the prescribed target stationarydistribution, $\hat \mu$. In this paper, we revisit the TSDP under a constrainton the support of $\Delta$, that is, on the set of non-zero entries of$\Delta$. This is particularly meaningful in practice since one cannottypically modify all entries of $G$. We first show how to construct a feasiblesolution $\hat G$ that has essentially the same support as the matrix $G$. Thenwe show how to compute globally optimal and sparse solutions using thecomponent-wise $\ell_1$ norm and linear optimization. We propose an efficientimplementation that relies on a column-generation approach which allows us tosolve sparse problems of size up to $10^5 \times 10^5$ in a few minutes. Weillustrate the proposed algorithms with several numerical experiments.

Assigning Stationary Distributions to Sparse Stochastic Matrices