Abstract
We develop a class of methods for minimizing a nondifferentiable function which is the maximum of a finite number of smooth functions. The methods proceed by solving iteratively quadratic programming problems to generate search directions. For efficiency the matrices in the quadratic programming problems are suggested to be updated in a variable metric way. By doing so, the methods possess many attractive features of variable metric methods and can be viewed as their natural extension to the nondifferentiable case. To avoid the difficulties of an exact line search, a practical stepsize procedure is also introduced. Under mild assumptions the resulting method converge globally.
Similar content being viewed by others
References
J.M. Danskin, “The theory of max—min, with applications”,SIAM Journal on Applied Mathematics 14 (1966) 641–664.
V.F. Dem'yanov and V.N. Malozemov,Introduction to minimax (John Wiley & Sons, New York, 1974).
J.E. Dennis Jr. and J.J. Moré, “Quasi-Newton methods, motivation and theory”,SIAM Review 19 (1977) 46–89.
S.P. Han, “Superlinearly convergent variable metric methods for general nonlinear programming”,Mathematical Programming 11 (1976) 263–282.
S.P. Han, “Dual variable metric methods for constrained optimization”,SIAM Journal on Control and Optimization 15 (1977) 546–565.
S.P. Han, “A global convergent method for nonlinear programming”,Journal of Optimization Theory and Applications 22 (1977) 297–309.
S.P. Han, “A hybrid method of nonlinear programming”, in O.L. Mangasarian, R.R. Meyer and S.M. Robinson, Eds.,Nonlinear Programming 3 (Academic Press, New York, 1978) 65–95.
S.P. Han, “Superlinear convergence of a minimax method”, Cornell University, Computer Science TR78-336 (1978).
C. Lemarechal, “An extension of Davidon methods to nondifferentiable problems”,Mathematical Programming Study 3 (1975) 95–109.
J.M. Ortega and W.C. Rheinboldt, Iterative solution of nonlinear equations in several variables (Academic Press, New York, 1970).
M.R. Osborne and G.A. Watson, “An algorithm for minimax approximation in the nonlinear case”,Computer Journal 12 (1969) 64–69.
M.J.D. Powell, “A fast algorithm for nonlinear constrained optimization calculations”, presented at the 1977 Dundee Conference on Numerical Analysis.
P. Wolfe, “A method of conjugate subgradients for minimizing nondifferentiable functions”,Mathematical Programming Study 3 (1975) 145–173.
Author information
Authors and Affiliations
Additional information
Research supported by National Science Foundation under grant number ENG 7903881.
Rights and permissions
About this article
Cite this article
Han, S.P. Variable metric methods for minimizing a class of nondifferentiable functions. Mathematical Programming 20, 1–13 (1981). https://doi.org/10.1007/BF01589328
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01589328