This communication is concerned with state- feedback stabilizability of discrete-time switched linear systems. Necessary and sufficient conditions for state-feedback exponential stabilizability are presented. It is shown that, a switched linear system is state-feedback exponentially stabilizable if and only if an associated sequence converges to zero. Equivalently, a switched linear system is state-feedback exponentially stabilizable if and only if a dynamic programming equation admits a solution of some kind. We also address the issue of testing the stabilizability of a given switched system by computing the elements of a new associated sequence of upper bounds for the elements of the previously mentioned sequence. These computations involve the solution of convex programming problems. The elements of both associated sequences are shown to be related via Lagrange duality. Numerical examples illustrate some of the results reported in the paper.