A Multi--Parameter Method for Nonlinear Least--Squares Approximation

. For discrete nonlinear least-squares approximation problems P m j=1 f 2 j (x) ! min for m smooth functions f j : IR n ! IR a numerical method is proposed which first minimizes each f j separately and then applies a penalty strategy to gradually force the different minimizers to coalesce. Though the auxiliary nonlinear least-squares method works on IR n1m , it is shown that the additional computational requirements consist of m 0 1 gradient evaluations plus O(m 1 n) operations. The application to discrete rational approximation is discussed and numerical examples are given. x1. Introduction Assume a nonlinear unconstrained mimimization problem f(x) ! min x2IR n ! (1) for a smooth function f : IR n ! IR to be given, and assume the problem (1) to have several local minimizers. We want to construct an algorithm that enhances the probability to find global minimizers. Led by the application to nonlinear least squares problems (which will be described in section 5) we assume..