One-parameter families of optimization problems: equality constraints. (English) Zbl 0556.90086

We introduce generalized critical points and discuss their relationship with other concepts of critical points [resp., stationary points]. Generalized critical points play an important role in parametric optimization. Under generic regularity conditions, we study the set of generalized critical points, in particular, the change of the Morse index. We focus our attention to problems with equality constraints only and provide an indication of how the present theory can be extended to problems with inequality constraints as well.


90C31 Sensitivity, stability, parametric optimization
Full Text: DOI


[1] McCormick, G. P.,Optimality Criteria in Nonlinear Programming, SIAM-AMS Proceedings, Vol. 9, pp. 27–38, 1976.
[2] Hettich, R., andJongen, H. Th.,On First-Order and Second-Order Conditions for Local Optima for Optimization Problems in Finite Dimensions, Methods of Operations Research, Vol. 23, pp. 82–97, 1977. · Zbl 0393.90078
[3] Milnor, J.,Morse Theory, Annals of Mathematic Studies, Study 51, Princeton University Press, Princeton, New Jersey, 1963.
[4] Marcus, M., andMinc, H.,A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston, Massachusetts, 1964. · Zbl 0126.02404
[5] Kojima, M.,Strongly Stable Stationary Solutions in Nonlinear Programs, Analysis and Computation of Fixed Points, Edited by S. M. Robinson, Academic Press, New York, New York, 1980. · Zbl 0478.90062
[6] Braess, D.,Morse Theorie für Berandete Mannigfaltigkeiten, Mathematische Annalen, Vol. 208, pp. 133–148, 1974. · Zbl 0263.58005
[7] Jongen, H. Th., Jonker, P. andTwilt, F.,Nonlinear Optimization in \(\mathbb{R}\) n ,I: Morse Theory, Chebyshev Approximation, Peter Lang Verlag, Frankfurt a.M., Germany 1983. · Zbl 0527.90064
[8] Hirsch, M. W.,Differential Topology, Springer Verlag, Berlin, Germany, 1976.
[9] Jongen, H. Th., Jonker, P., andTwilt, F.,Nonlinear Optimization in \(\mathbb{R}\) n ,II: Transversality, Flows, Parametric Aspects (to appear). · Zbl 0611.90071
[10] Lu, Y. C.,Singularity Theory and an Introduction to Catastrophe Theory, Universitext, Springer Verlag, Berlin, Germany, 1976. · Zbl 0354.58008
[11] Jongen, H. Th., Jonker, P., andTwilt, F.,On One-Parameter Families of Sets Defined by (In) Equality Constraints, Nieuw Archief voor Wiskunde, Vol. 30, pp. 307–322, 1982. · Zbl 0518.58032
[12] Jongen, H. Th., Jonker, P., andTwilt, F.,On Deformation in Optimization, Methods of Operations Research, Vol. 37, pp. 171–184, 1980. · Zbl 0459.90075
[13] Jongen, H. Th., Jonker, P., andTwilt, F.,Critical Sets in Parametric Optimization, Mathematical Programming (to appear). · Zbl 0599.90114
[14] Kojima, M., andHirabayashi, R.,Some Results on the Strong Stability in Nonlinear Programs, Tokyo Institute of Technology, Department of Management Science and Engineering, Technical Report No. 4, 1980.
[15] Kojima, M., andHirabayashi, R.,Continuous Deformation of Nonlinear Programs, Mathematical Programming Study, Vol. 21, pp. 150–198, 1984. · Zbl 0569.90074
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.