Karush–Kuhn–Tucker conditions for rank-deficient nonlinear least-square problems with rank-deficient nonlinear constraints

In nonlinear least-square problems with nonlinear constraints, the norm of a nonlinear vector function is to be minimized subject to the nonlinear equality constraints. This problem is ill posed if the first-order KKT conditions do not define a locally unique solution. We show that the problem is ill posed if either the Jacobian of the first derivative of the objective or the Jacobian of a function of the constraints is rank-deficient (i.e., not of full rank) in a neighbourhood of a solution satisfying the first-order KKT conditions. Either of these ill-posed cases makes it impossible to use a standard Gauss–Newton method. Therefore, we formulate a constrained least-norm problem that can be used when either of these ill-posed cases occur. By using the constant-rank theorem, we derive the necessary and sufficient conditions for a local minimum of this minimum-norm problem. The results given here are crucial for deriving methods solving the rank-deficient problem.