This note describes a model predictive control (MPC) formulation for discrete-time linear systems with hard constraints on control and state variables, under the assumption that the solution of the associated quadratic program is neither optimal nor satisfies the inequality constraints. This is common in embedded control applications, for which real-time constraints and limited computing resources dictate restrictions on the possible number of on-line iterations that can be performed within a sampling period. The proposed approach is rather general, in that it does not refer to a particular optimization algorithm, and is based on the definition of an alternative MPC problem that we assume can only be solved within bounded levels of suboptimality, and violation of the inequality constraints. By showing that the inexact solution is a feasible suboptimal one for the original problem, asymptotic or exponential stability is guaranteed for the closed-loop system. Based on the above general results, we focus on a specific dual accelerated gradient-projection method to obtain a stabilizing MPC law that only requires a predetermined maximum number of on-line iterations.