Managing computational effort (CPU time, memory, interfacing) is a major issue in design optimization, due to the cost of the high fidelity numerical simulations (finite elements, finite volumes, etc.) involved. In order to decrease the overall cost of the optimization process, reduced-order models such as Proper Orthogonal Decomposition (POD) are an economical and efficient option. However, truncating the POD basis yields an error in the calculation of the global values used as performance objectives and constraints which in turn affects the optimization results. This paper proposes novel constrained versions of Proper Orthogonal Decomposition that produce an alternative orthonormal basis, which is then successfully applied first to a 1D test-case with a quadratic constraint and next to an industrial example with both linear and quadratic constraints: the multi-objective shape optimization of an air-conditioning duct.