Model predictive control and identification (MPCI) is an adaptive MPC scheme that employs the persistent excitation condition to generate non-convex constraints on process inputs, in addition to conventional constraints of on-line optimization of MPC. This results in a non-convex problem, which has to be solved at each time instant k. In this work we rigorously show that a locally optimal solution of the above problem can be obtained by the successive semidefinite programming algorithm. We then develop a deterministic branch-and-bound approach, to obtain the global minimum of the MPCI optimization problem. Simulation results are presented to demonstrate the applicability of the proposed MPCI algorithms to small-scale systems.