KKT conditions for rank-deficient nonlinear least-square problems with rank-deficient nonlinear constraints

In nonlinear least-square problems with nonlinear constraints, the function (1/2) // f(2)(x) // (2)(2), where f(2) is a nonlinear vector function, is to be minimized subject to the nonlinear constraints fi (x) = 0. This probl em is ill-posed if the first-order KKT conditions do not define a locally u nique solution. We show that the problem is ill-posed if either the Jacobia n of f(1) or the Jacobian of J is rank-deficient (i.e., not of full rank) i n a neighborhood of a solution satisfying the first-order KKT conditions. E ither of these ill-posed cases makes it impossible to use a standard Gauss- Newton method. Therefore, we formulate a constrained least-norm problem tha t can be used when either of these ill-posed cases occur. By using the cons tant-rank theorem, we derive the necessary and sufficient conditions for a local minimum of this minimum-norm problem. The results given here are cruc ial for deriving methods solving the rank-deficient problem.