An Algorithm for Constrained Non-Linear Least-Squares

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Abstract
An algorithm is presented for finding a constrained stationary point of a non-linear least squares problem when the constraints are upper bounds, lower bounds, and a set x1 ≤ x2 ≤…≤ xk. The method is based on the unconstrained Levenberg-Marquardt algorithm, reducing at each iteration to an unconstrained least squares problem. Also incorporated is a model-prediction of the desirability of leaving an active constraint at each interation, so that it is not necessary to locate an approximate local minimum in a manifold before investigating whether to alter the manifold. A new method of control for the damping parameter is presented, and three numerical examples given covering the inversion of a Fredholm integral equation and spline and exponential data fitting.

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