This paper deals with the optimal selection of linear measurement combinations as controlled variables, c = Hy. The objective is to achieve "self-optimizing control", which is when fixing the controlled variables c indirectly gives near-optimal steady-state operation with a small loss. The millspace method of Alstad and Skogestad [V. Alstad, S. Skogestad, Null space method for selecting optimal measurement combinations as controlled variables, Ind. Eng. Chem. Res. 46 (3) (2007) 846-853] focuses on minimizing the loss caused by disturbances. We here provide an explicit expression for H for the case where the objective is to minimize the combined loss for disturbances and measurement errors. In addition, we extend the nullspace method to cases with extra measurements by using the extra degrees of freedom to minimize the loss caused by measurement errors. Finally, the results are interpreted more generally as deriving linear invariants for quadratic optimization problems. (C) 2008 Elsevier Ltd. All rights reserved.