Increase of drilling safety and reduction of drilling operation costs, especially improvement of drilling efficiency, are two important considerations in the oil and gas industry. The rate of penetration (ROP, alternatively called as drilling speed) is a critical drilling parameter to evaluate and improve drilling safety and efficiency. ROP estimation has an important role in drilling optimization as well as interpretation of all stages of the well life cycle. In this paper, we use a moving horizon estimation (MHE) method to estimate ROP as well as other drilling parameters. In the MHE formulation the states are estimated by a forward simulation with a pre-estimating observer. Moreover, it considers the constraints of states/outputs in the MHE problem. It is shown that the estimation error is with input-to-state stability. Furthermore, the ROP optimization (to achieve minimum drilling cost/drilling energy) concerning with the efficient hole cleaning condition and downhole environmental stability is presented. The performance of the methodology is demonstrated by one case study.