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Convergence of discrete approximations for constrained minimization

Published online by Cambridge University Press:  17 February 2009

B. D. Craven
B. D. Craven
Affiliation:
Mathematics Department, University of Melbourne, Parkville, Victoria 3052, Australia
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Abstract

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If a constrained minimization problem, under Lipschitz or uniformly continuous hypotheses on the functions, has a strict local minimum, then a small perturbation of the functions leads to a minimum of the perturbed problem, close to the unperturbed minimum. Conditions are given for the perturbed minimum point to be a Lipschitz function of a perturbation parameter. This is used to study convergence rate for a problem of continuous programming, when the variable is approximated by step-functions. Similar conclusions apply to computation of optimal control problems, approximating the control function by step-functions.

Type
Research Article
Information
The ANZIAM Journal , Volume 36 , Issue 1 , July 1994 , pp. 50 - 59
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Alt, W., “Stability of solutions for a class of nonlinear cone constrained optimization problems”, Numer. Fund. Anal. Appl. 10 (1989) 1053–1064 and 1064–1076.Google Scholar
[2]Alt, W., Stability of solutions and the Lagrange-Newton method for nonlinear optimization and optimal control problems (Habilitationsschrift, Universität Bayreuth, 1990).Google Scholar
[3]Craven, B. D., “Perturbed minimization, with constraints adjoined or deleted”, Optimization 14 (1983) 23–26.Google Scholar
[4]Craven, B. D., “Nondifferentiable optimization by smooth approximations”, Optimization 17 (1986) 3–17.Google Scholar
[5]Craven, B. D. and Janin, R., “On stability in constrained optimization”, submitted for publication, 1992.Google Scholar
[6]Jennings, L. S., Fisher, M. E., Teo, K. L. and Goh, C. J., MISER3, Optimal Control Software, Theory and User Manual (1990).Google Scholar
[7]Lempio, F. and Maurer, H., “Differential stability in infinite-dimensional nonlinear programming”, Appl. Math. Optim. 6 (1980) 139–152.CrossRefGoogle Scholar
[8]Robinson, S. M., “Stability theory for systems of inequalities, Part II: Differentiable nonlinear systems”, SIAMJ. Numer. Anal. 13 (1976) 497–513.Google Scholar
[9]Teo, K. L. and Goh, C. J., “A computational method for combined optimal parameter selection and optimal control problems with general constraints”, J. Austral. Math. Soc. Ser. B 30 (1989) 350–364.Google Scholar
[10]Teo, K. L., Goh, C. J. and Wong, K. H., A unified computational approach to optimal control problems (Longman Scientific and Technical, 1991).Google Scholar
[11]Teo, K. L. and Womersley, R. S., “A control parametrization algorithm for optimal control problems involving linear systems and linear terminal inequality constraints”, Numer. Fund. Anal. Opt. 6 (1983) 291–313.Google Scholar
[12]Yen, N. D., “Stability of the solution set of perturbed nonsmooth inequality systems”, J. Optim. Theor. Appl., to appear.Google Scholar