Abstract
A trust region algorithm is proposed for minimizing the nonsmooth composite functionF(x) = h(f(x)), wheref is smooth andh is convex. The algorithm employs a smoothing function, which is closely related to Fletcher's exact differentiable penalty functions. Global and local convergence results are given, considering convergence to a strongly unique minimizer and to a minimizer satisfying second order sufficiency conditions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T. Bannert, “Ein Trust-Region-Verfahren basierend auf der Glättung nichtdifferenzierbarer Funktionen,” Ph.D. Thesis, Georg-August-Universität Göttingen (Göttingen, 1993).
F.H. Clarke,Optimization and Nonsmooth Analysis (John Wiley & Sons, New York, 1983).
R. Fletcher, “A class of methods for nonlinear programming with termination and convergence properties,” in: J. Abadie, ed.,Integer and Nonlinear Programming (North-Holland, Amsterdam—London, 1970) pp. 157–175.
R. Fletcher, “A class of nonlinear programming III: Rates of convergence,” in: F. Lootsma, ed.,Numerical Methods for Nonlinear Optimization (Academic Press, London—New York, 1972) pp. 371–381.
R. Fletcher, “An exact penalty function for nonlinear programming with inequalities,”Mathematical Programming 5 (1973) 129–150.
R. Fletcher, “A model algorithm for composite nondifferentiable optimization problems,”Mathematical Programming Study 17 (1982) 67–76.
R. Fletcher, “Second order corrections for non-differentiable optimization,” in: G.A. Watson, ed.,Numerical Analysis Proceedings (Springer, Berlin, 1982) pp. 85–114.
R. Fletcher,Practical Methods of Optimization (John Wiley & Sons, New York, 1987).
K. Madsen, “An algorithm for minimax-solution of overdetermined systems of nonlinear equations,”Journal of the Institute of Mathematics and its Applications 16 (1975) 321–328.
M.J.D. Powell and Y. Yuan, “A trust region algorithm for equality constrained optimization,”Mathematical Programming 49 (1990) 189–211.
R.S. Womersley, “Local properties of algorithms for minimizing nonsmooth composite functions,”Mathematical Programming 32 (1985) 69–89.
Y. Yuan, “An example for only linear convergence of trust region algorithms for non-smooth optimization,”IMA Journal of Numerical Analysis 4 (1984) 327–335.
Y. Yuan, “Conditions for convergence of trust region algorithms for nonsmooth optimization,”Mathematical Programming 31 (1985) 220–228.
Y. Yuan, “On the superlinear convergence of a trust region algorithm for non-smooth optimization,”Mathematical Programming 31 (1985) 269–285.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bannert, T. A trust region algorithm for nonsmooth optimization. Mathematical Programming 67, 247–264 (1994). https://doi.org/10.1007/BF01582223
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01582223