  We propose a Model Predictive Control (MPC) with a single-step prediction
horizon to solve infinite horizon optimal control problems with the expected
sum of convex stage costs for constrained linear uncertain systems. The
proposed method relies on two techniques. First, we estimate the expected
values of the convex costs using a computationally tractable approximation,
achieved by sampling across the space of disturbances. Second, we implement a
data-driven approach to approximate the optimal value function and its
corresponding domain, through systematic exploration of the system's state
space. These estimates are subsequently used as the terminal cost and terminal
set within the proposed MPC. We prove recursive feasibility, robust constraint
satisfaction, and convergence in probability to the target set. Furthermore, we
prove that the estimated value function converges to the optimal value function
in a local region. The effectiveness of the proposed MPC is illustrated with
detailed numerical simulations and comparisons with a value iteration method
and a Learning MPC that minimizes a certainty equivalent cost.
