Summary: This paper presents some solution approaches to the problem of optimal energy-regenerative model predictive control for linear systems subject to stability and/or dissipativity constraints, as well as hard constraints on the state and control vectors. The problem is generally non-convex in the objective and some of the constraints, thereby resulting in a non-convex optimization problem to be solved at each time step. Multiple extended convex relaxation approaches are considered. As a result, a more conservative semi-definite programming problem is proposed to be solved at each time step. The feasibility and stability of the resulting closed-loop system are also examined. The approaches are validated using a numerical example of maximizing energy regeneration from a single degree of freedom vibrating system subject to a level-set constraint on some performance metric characterizing the quality of vibration isolation achieved by the system. The constraint is described in terms of an upper bound on the \(\mathcal L_2\)-gain of the system from the input to a vector of appropriately selected system outputs.