Abstract
An efficient algorithm for solving nonlinear programs with noisy equality constraints is introduced and analyzed. The unknown exact constraints are replaced by surrogates based on the bundle idea, a well-known strategy from nonsmooth optimization. This concept allows us to perform a fast computation of the surrogates by solving simple quadratic optimization problems, control the memory needed by the algorithm, and prove the differentiability properties of the surrogate functions. The latter aspect allows us to invoke a sequential quadratic programming method. The overall algorithm is of the quasi-Newton type. Besides convergence theorems, qualification results are given and numerical test runs are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
WINFIELD, D., Function Minimization by Interpolation in a Data Table, Journal of the Institute of Mathematics and Its Applications, Vol. 12, pp. 339–347, 1973.
BORTZ, D. M., and KELLEY, C. T., The Simplex Gradient and Noisy Optimization Problems, Computational Methods in Optimal Design and Control, Edited by J. T. Borggaard, J. Burns, E. Cliff, and S. Schreck, Progress in Systems and Control Theory, Birkhäuser, Boston, Massachusetts, Vol. 24, pp. 77–90, 1998.
ELSTER, C., and NEUMAIER, A., A Grid Algorithm for Bound-Constrained Optimization of Noisy Functions, IMA Journal of Numerical Analysis, Vol. 15, pp. 585–608, 1995.
ELSTER, C., and NEUMAIER, A., A Trust-Region Method for the Optimization of Noisy Functions, Computing, Vol. 58, pp. 31–46, 1997.
GLAD, T., and GOLDSTEIN, A., Optimization of Functions Whose Values Are Subjected to Small Errors, BIT, Vol. 17, pp. 160–169, 1977.
NELDER, J. A., and MEAD, R., A Simplex Method for Function Minimization, Computer Journal, Vol. 7, pp. 308–313, 1965.
TORCZON, V., On the Convergence of the Multidirectional Search Algorithm, SIAM Journal on Optimization, Vol. 1, pp. 123–145, 1991.
TORCZON, V., On the Convergence of Pattern Search Algorithms, SIAM Journal on Optimization, Vol. 7, pp. 1–25, 1997.
KELLEY, C. T., and SACHS, E., Truncated Newton Methods for Optimization with Inexact Functions and Gradients, Technical Report CRSC-TR99–20, North Carolina State University, 1999.
HINTERMüLLER, M., An Algorithm for Solving Nonlinear Programs with Noisy Inequality Constraints, Nonlinear Optimization and Related Topics, Edited by G. Di Pillo and F. Giannessi, Kluwer Academic Publishers, Dordrecht, Holland, pp. 143–168, 1999.
POWELL, M. J. D., Variable Metric Methods for Constrained Optimization, Mathematical Programming: The State of the Art (Bonn, 1982), Springer Verlag, Berlin, Germany, pp. 288–311, 1983.
DESAI, M., and ITO, K. Optimal Controls of Navier-Stokes Equations, SIAM Journal on Control and Optimization, Vol. 5, pp. 1428–1446, 1994.
BARCLAY, A., GILL, P. E., and ROSEN, J. B., SQP Methods and Their Application to Numerical Optimal Control, Variational Calculus, Optimal Control, and Applications (Trassenheide, 1996), International Series of Numerical Mathematics, Birkhäuser, Boston, Massachusetts, Vol. 24, pp. 207–222, 1998.
DENNIS, J. E., HEINKENSCHLOSS, M., and VICENTE, L. N., Trust-Region Interior-Point SQP Algorithms for a Class of Nonlinear Programming Problems, SIAM Journal on Control and Optimization, Vol. 36, pp. 1750–1794, 1998.
KIWIEL, K. C., Proximity Control in Bundle Methods for Convex Nondifferentiable Minimization, Mathematical Programming, Vol. 46, pp. 105–122, 1990.
ZOWE, J., Nondifferentiable Optimization, Computational Mathematical Programming, Edited by K. Schittkowski, Series F: Computer and System Sciences, Springer Verlag, New York, NY, Vol. 15, pp. 323–356, 1985.
SHAPIRO, A., Sensitivity Analysis of Nonlinear Programs and Differentiability Properties of Metric Projections, SIAM Journal on Control and Optimization, Vol. 26, pp. 628–645, 1988.
WRIGHT, S. J., Primal-Dual Interior-Point Methods, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1997.
ZHANG, Y., Solving Large-Scale Linear Programs by Interior-Point Methods under the MATLAB Environment, Technical Report TR96–01, Department of Mathematics, University of Maryland, Baltimore, Maryland, 1996.
KIWIEL, K. C., A Method for Solving Certain Quadratic Programming Problems Arising in Nonsmooth Optimization, IMA Journal of Numerical Analysis, Vol. 6, pp. 137–152, 1986.
KIWIEL, K. C., A Dual Method for Certain Positive-Semidefinite Quadratic Programming Problems, SIAM Journal on Scientific and Statistical Computing, Vol. 10, pp. 175–186, 1989.
FLETCHER, R., Practical Methods of Optimization, Vol 2, John Wiley and Sons, New York, NY, 1980.
ROBINSON, S. M., Generalized Equations and Their Solutions, Part 2: Applications to Nonlinear Programming, Mathematical Programming Study, Vol. 19, pp. 200–221, 1982.
BONNANS, J. F., et al., Optimization Numérique, Mathématiques et Applications, Springer Verlag, Berlin, Germany, Vol. 27, 1997.
SCHITTKOWSKI, K., and HOCK, W., Test Examples for Nonlinear Programming Codes, Springer Verlag, New York, NY, 1981.
SCHITTKOWSKI, K., More Test Examples for Nonlinear Programming Codes, Springer Verlag, New York, NY, 1987.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hintermüller, M. Solving Nonlinear Programming Problems with Noisy Function Values and Noisy Gradients. Journal of Optimization Theory and Applications 114, 133–169 (2002). https://doi.org/10.1023/A:1015416221909
Issue Date:
DOI: https://doi.org/10.1023/A:1015416221909