We consider the problem of discounted optimal state-feedback regulation for general unknown deterministic discrete-time systems. It is well known that open-loop instability of systems, non-quadratic cost functions and complex nonlinear dynamics, as well as the on-policy behavior of many reinforcement learning (RL) algorithms, make the design of model-free optimal adaptive controllers a challenging task... We depart from commonly used least-squares and neural network approximation methods in conventional model-free control theory, and propose a novel family of data-driven optimization algorithms based on linear programming, off-policy Q-learning and randomized experience replay. We develop both policy iteration (PI) and value iteration (VI) methods to compute an approximate optimal feedback controller with high precision and without the knowledge of a system model and stage cost function. Simulation studies confirm the effectiveness of the proposed methods. (read more)