We study the problem of computing the maximal admissible positively invariant set for discrete time switching affine systems subject to basic semi-algebraic constraints. First, we obtain inner ϵ-approximations of the minimal invariant set. Second, following recent results for switching linear systems (Athanasopoulos and Jungers, 2016), we apply an algebraic lifting on the system and obtain a polyhedral representation of the constraint set. Working on this lifted state space offers two distinct advantages, namely (i) we can verify inclusion of an epsilon-inflation of the minimal invariant set in the constraint set and (ii) under proper assumptions, we can characterize and compute the maximal admissible invariant set, which is also a basic semi-algebraic set. Consequently, we are able to identify and recover admissible invariant sets for switching affine systems even when only non-convex invariant sets exist. The underlying algorithms involve only linear operations and convex hull computations.