Boundedness of the inverse of a regularized Jacobian matrix in constrained optimization and applications

This short paper describes a boundedness property of the inverse of a regularized Jacobian matrix that arises in optimization algorithms for solving constrained problems. The regularization considered here is to overcome a rank deficiency of the Jacobian matrix of constraints. We show that the norm of the inverse of the regularized matrix is locally bounded by a coefficient inversely proportional to the regularization parameter. We also show how this result can be used for the local convergence analysis of algorithms without a constraint qualification hypothesis.