QUASI-CONVEX SETS AND SIZE X CURVATURE CONDITION, APPLICATION TO NONLINEAR INVERSION

We define a family of sets of a Hilbert space ("quasi-convex sets") on which a generalization of the usual theory of projection on convex sets can be defined (existence, uniqueness, and stability of the projection of all points of some neighborhood of the set). We then give a constructive sufficient condition, called the size x curvature condition, for a set D to be quasi-convex, which involves radii of curvatures of curves lying on the set D. Finally, we use the above result for the study of nonlinear least-squares problems, as they appear in parameter estimation, for which we give a sufficient condition ensuring existence, uniqueness, and stability.