Summary
Strong uniqueness has proved to be an important condition in demonstrating the second order convergence of the generalised Gauss-Newton method for discrete nonlinear approximation problems [4]. Here we compare strong uniqueness with the multiplier condition which has also been used for this purpose. We describe strong uniqueness in terms of the local geometry of the unit ball and properties of the problem functions at the minimum point. When the norm is polyhedral we are able to give necessary and sufficient conditions for the second order convergence of the generalised Gauss-Newton algorithm.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Anderson, D.H., Osborne, M.R.: Discrete linear approximation problems in polyhedral norms. Numer. Math.26, 179–189 (1976)
Anderson, D.H., Osborne, M.R.: Discrete nonlinear approximation problems in polyhedral norms. Numer. Math28, 143–156 (1977)
Cheney, E.W.: Introduction to approximation theory. New York: McGraw-Hill 1966
Crome, L.: Strong uniqueness. A far reaching criterion for the convergence analysis of iterative procedures. Numer. Math.,29, 179–194 (1978)
Fiacco, A.V., McCormick, G.P.: Nonlinear programming: sequential unconstrained minimization techniques New York: Wiley 1968
Jittorntrum, K.; Sequential algorithms in nonlinear programming, Ph.D. Thesis, Australian National University, 1978
Luenberger, D.G.: Optimisation by vector space methods. New York: Wiley 1968
Osborne, M.R.: An algorithm for discrete, nonlinear, best approximation problems. In: Numerische Methoden der Approximationstheorie, Band 1. L. Collatz, G. Meinardus, (Hrsg.). Basel-Stuttgart: Birkhäuser-Verlag 1972
Osborne, M.R., Watson, G.A.: Nonlinear approximation in vector norms. In: numerical analysis, G.A. Watson, (ed.) Lecture Notes in Mathematics No. 630, pp. 117–133. Berlin-Heidelberg-New York: Springer 1978
Rockafellar, R.T.: Convex analysis, Princeton: Princeton University Press, 1970
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jittorntrum, K., Osborne, M.R. Strong uniqueness and second order convergence in nonlinear discrete approximation. Numer. Math. 34, 439–455 (1980). https://doi.org/10.1007/BF01403680
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01403680