On the uniqueness of local minima for general abstract nonlinear least-squares problems

The effectiveness of the inversion of a mapping phi defined on a set C by nonlinear least-squares techniques relies on, among other things, the uniqueness of local minima of the least-squares criterion, which ensures that the numerical optimisation algorithm (if they do) converges towards the global minimum of the least-squares functional. The author defines a number y depending only on C and phi which, if the size of phi (C) is not too large with respect to its curvature, is strictly positive, thus yielding the uniqueness of all local minima having a value smaller than y. The condition y>0 requires neither convexity of C nor any monotonic property of phi , but involves the computation of an infimum over delta C* delta C of first and second derivatives of phi . Numerical applications to the estimation of two parameters in a parabolic equation are given.