In this paper we present a rigorous method for the construction of enhanced Proper Orthogonal Decomposition (POD) projection bases for the development of efficient Reduced Order Models (ROM). The resulting ROMs are seen to exactly interpolate global quantities by design, such as the objective function(s) and nonlinear constraints involved in the optimization problem, thus narrowing the search space by limiting the number of constraints that need to be explicitly included in the statement of the optimization problem. We decompose the basis into two subsets of orthogonal vectors, one for the representation of constraints and the other one, in a complementary space, for the minimization of the projection errors. An explicit algorithm is presented for the case of linear objective functions. The proposed method is then implemented within a bi-level ROM and is illustrated with an application to the multi-objective shape optimization of a car engine intake port for two competing objectives: CO2 emissions and engine power. We show that optimization using the proposed method produces Pareto dominant and realistic solutions for the flow fields within the combustion chamber, providing further insight into the flow properties.