Abstract
In this paper, continuity properties of the extremal value function and the solution function are studied for general optimization problems with perturbations in the objective function and the constraints. A classical stability condition is extended and compared with constraint qualification conditions.
Similar content being viewed by others
References
Krabs, W.,Zur Stetigen Abhängigkeit des Extremwertes eines Konvexen Optimierungsproblems von der Stetigen Änderung des Problems, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 52, pp. 359–368, 1972.
Esser, H.,Über die Stetigkeit des Extremalwertes Nichtkonvexer Optimierung-sprobleme mit einer Anwendung auf die Diskretisierung von Kontrollproblemen, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 52, pp. 535–542, 1972.
Gauvin, J., andTolle, J. W.,Differential Stability in Nonlinear Programming, SIAM Journal on Control and Optimization, Vol. 15, pp. 294–311, 1977.
Rockafellar, R. T.,Duality and Stability in Extremum Problems Involving Convex Functions, Pacific Journal of Mathematics, Vol. 21, pp. 167–187, 1967.
Evans, J. P., andGould, F. J.,Stability in Nonlinear Programming, Operations Research, Vol. 18, pp. 107–118, 1970.
Greenberg, H. J., andPierskalla, W. P.,Extensions of the Evans-Gould Stability Theorem for Mathematical Programs, Operations Research, Vol. 20, pp. 143–153, 1972.
Lempio, F., andMaurer, H.,Differential Perturbations of Infinite Optimization Problems, Reprint, 1977.
Kirsch, A., Warth, W., andWerner, J.,Notwendige Optimalitäts-bedingungen und Ihre Anwendung, Springer-Verlag, Berlin, Germany, 1978.
Luenberger, D.,Optimization by Vector Space Methods, John Wiley and Sons, New York, New York, 1969.
Reindlmeier, J.,Zur Stetigen Parameterabhängigkeit der Menge der Minimalpunkte bei MinMax-Aufgaben, University of Göttingen, Göttingen, Germany, PhD Thesis, 1975.
Robinson, S. M.,Stability Theory for Systems of Inequalities, Part 2, Differentiable Nonlinear Systems, 2, SIAM Journal on Numerical Analysis, Vol. 13, pp. 497–573, 1976.
Cullum, J.,Discrete Approximations to Continuous Optimal Control Problems, SIAM Journal on Control and Optimization, Vol. 7, pp. 32–49, 1969.
Cullum, J.,Perturbations of Optimal Control Problems, SIAM Journal on Control and Optimization, Vol. 4, pp. 473–487, 1966.
Budak, B. M., Berkovich, E. M., andSolov'eva, E. N.,Difference Approximations in Optimal Control Problems, SIAM Journal on Control and Optimizations Vol. 7, pp. 18–31, 1969.
Hogan, W. W.,Point-to-Set Maps in Mathematical Programming, SIAM Review, Vol. 15, pp. 591–603, 1973.
Author information
Authors and Affiliations
Additional information
Communicated by D. G. Luenberger
Rights and permissions
About this article
Cite this article
Kirsch, A. Continuous perturbations of infinite optimization problems. J Optim Theory Appl 32, 171–182 (1980). https://doi.org/10.1007/BF00934722
Issue Date:
DOI: https://doi.org/10.1007/BF00934722