Influence of Equality Constraints on Ill-conditioned Problems and Constrained Regularization Method

The accuracy of least square estimation can be improved and the rank deficient problem can be eliminated by using equality constraints properly. However, the influence of equality constraints in an ill-conditioned problem is not clear. Therefore, a method to transform the equality-constrained ill-posed to unconstrained problem by eliminating a part of the parameters is proposed in this paper. We firstly analyzed the influence of equality constraints on an ill-posed problem. Subsequently, the new system with equality constraints was verified by means of an example as either ill-conditioned or not depending on the original design matrix and equality constraints. We finally propose a diagnosis-regularization two-step approach to solve the equality-constrained ill-posed problem and validate its feasibility with a simulated data experiment.