A continuation method for solving separable nonlinear least squares problems

AbstractGiven the data (xi, yi), i=1, 2, …, n, the problem is to find the values of the linear and nonlinear parameters â and b̂ which minimize the nonlinear functional |F(b)a−y|22 over a ϵ Rp, b ϵ Rq, where F ϵ Rn×p is a variable matrix and assumed to be of full rank, and y ϵ Rn is a constant vector.In this paper, we present a method for solving this problem by imbedding it into a one-parameter family of problems and by following its solution path using a predictor-corrector algorithm. In the course of iterations, the original problem containing p+q+1 variables is transformed into a problem with q+1 nonlinear variables by taking the separable structure of the problem into account. By doing so, the method reduces to solving a series of equations of smaller size and a considerable saving in the storage is obtained.Results of numerical experiments are reported to demonstrate the effectiveness of the proposed method