Results concerning the output regulation problem for over-actuated linear systems are presented in this paper. The focus is on the characterization of the solution of the full-information regulator problem for systems which are right-invertible (but not left-invertible) and the input operator is injective. The intrinsic redundancy in the plant model is exploited by parameterizing all solutions of the regulator equations and performing a static or dynamic optimization on the space of solutions. This approach effectively shapes the non-unique steady-state of the system so that the long-term behavior optimizes a given performance index. In particular, nonlinear cost functions that account for constraints on the inputs are considered, within the general form of a hybrid system assumed for the allocation mechanism. An example is given to illustrate the proposed methodology. (C) 2014 Elsevier Ltd. All rights reserved.