This paper proposes a novel Bayesian approach to the problem of variable selection and shrinkage in high dimensional sparse non–negative linear regression models. The regularisation method is an extension of the LASSO which has been re- cently cast in a Bayesian framework by Park and Casella (2008). Moreover, to deal with the additional problem of variable selection we propose a Stochastic Search Variable Selection (SSVS) method that relies on a dirac spik–and–slab prior where the slab component induces the sparse non–negative regularisation. The methodol- ogy is then applied to the problem of passive index tracking of large dimensional index in stock markets without short sales.