## A new sufficient condition for the well-posedness of non-linear least square problems arising in identification and control.(English)Zbl 0702.93070

Analysis and optimization of systems, Proc. 9th Int. Conf., Antibes/Fr. 1990, Lect. Notes Control Inf. Sci. 144, 452-463 (1990).
Summary: [For the entire collection see Zbl 0699.00041.]
We show how simple 1-D geometrical calculations (but along all maximal segments of the parameter or control set!) can be used to establish the wellposedness of a nonlinear least-square (NLLS) problem and the absence of local minima in the corresponding error function. These sufficient conditions, which are shown to be sharp by elementary examples, are based on the use of the recently developed “size $$\times curvature''$$ conditions [see the author, “New size $$\times curvature$$ conditions for strict quasiconvexity of sets”, INRIA Report (to appear)] for proving that the output set is strictly quasiconvex. The use of this geometrical theory as a numerical or theoretical tool is discussed. Finally, application to regularized NLLS problem is shown to give new information on the choice of the regularizing parameter.

### MSC:

 93E24 Least squares and related methods for stochastic control systems 93B30 System identification 65D10 Numerical smoothing, curve fitting

### Keywords:

nonlinear least-square (NLLS); regularized NLLS

Zbl 0699.00041