KKT Conditions for Rank-Deficient Nonlinear Least-Square Problems with Rank-Deficient Nonlinear Constraints

M. Gulliksson
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M. Gulliksson: Umeå University

Journal of Optimization Theory and Applications, 1999, vol. 100, issue 1, No 7, 145-160

Abstract: Abstract In nonlinear least-square problems with nonlinear constraints, the function $$\left. {(1/2)} \right\|\left. {f_2 (x)} \right\|_2^2$$ , where f 2 is a nonlinear vector function, is to be minimized subject to the nonlinear constraints f 1(x)=0. 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 f 1 or the Jacobian of J is rank-deficient (i.e., not of full rank) in a neighborhood 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.

Keywords: Nonlinear least squares; optimization; regularization; KKT conditions; rank-deficient nonlinear constraints; rank-deficient nonlinear least-square problems (search for similar items in EconPapers)
Date: 1999
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DOI: 10.1023/A:1021721132282

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