In this paper we consider the quadratic optimal control problem of a discrete-time Markovian jump linear system, subject to constraints on the state and control variables. It is desired to find a state feedback controller, which may also depend on the jump variable, that minimizes a quadratic cost and satisfies some upper bounds on the norms of some random variables, related to the state and control variables of the system. The transition probability of the Markov chain and initial condition of the system may belong to appropriate convex sets. We obtain an approximation for the optimal solution of this problem in terms of linear matrices inequalities, so that convex programming can be used for numerical calculations. Examples are presented to illustrate the usefulness of the developed results.