Critical sets in parametric optimization. (English) Zbl 0599.90114

The authors consider the nonlinear parametric programming problem. They introduce the concept of a generalized critical point for the nonlinear programming problem as well as for the nonlinear parametric programming problem. The set of generalized critical points for the parametric programming problem is divided into five types and each type is studied separately. In the end of the paper the set of Kuhn-Tucker points as a subset of the set of generalized critical points is considered in more detail.
Reviewer: E.Tamm


90C31 Sensitivity, stability, parametric optimization
90C30 Nonlinear programming
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[1] E.L. Allgower and K. Georg, ”Predictor-corrector and simplicial methods for approximating fixed points and zero points of nonlinear mappings”, in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical programming, the state of the art (Springer-Verlag, Berlin, 1983) pp. 15–56. · Zbl 0541.65032
[2] V.I. Arnol’d, A.N. Varchenko and S.M. Gusein-Zade,Singularities of differentiable maps (Nayka, Moscow, 1982) (in Russian).
[3] A.V. Fiacco,Introduction to sensitivity and stability analysis in nonlinear programming (Academic Press, New York, 1983). · Zbl 0543.90075
[4] A.V. Fiacco, ed., ”Sensitivity, stability and parametric analysis”,Mathematical Programming Study 21 (1984). · Zbl 0602.00009
[5] J. Guddat and K. Wendler, ”On dialogue-algorithms for linear and nonlinear vector optimization from the point of view of parametric programming”, in: M. Grauer and A.P. Wierzbicki, eds.,Interactive decision analysis proceedings, Laxenburg, Austria 1983, Lecture Notes in Economics and Mathematical Systems 229 (Springer-Verlag, Berlin, 1984).
[6] R. Hettich and P. Zencke,Numerische Methoden der Approximation und semi-infiniten Optimierung (Teubner Studienbücher, Stuttgart, 1982). · Zbl 0481.65033
[7] M.W. Hirsch,Differential topology (Springer-Verlag, Berlin, 1976). · Zbl 0356.57001
[8] H.Th. Jongen, P. Jonker and F. Twilt, ”On deformation in optimization”,Methods of Operations Research 37 (1980) 171–184. · Zbl 0459.90075
[9] H.Th. Jongen, P. Jonker and F. Twilt, ”On one-parameter families of sets defined by (in)equality constraints”,Nieuw Archief voor Wiskunde (3) 30 (1982) 307–322. · Zbl 0518.58032
[10] H.Th. Jongen, P. Jonker and F. Twilt,Nonlinear optimization in \(\mathbb{R}\) n, I. Morse theory, Chebyshev approximation, Methoden und Verfahren der mathematischen Physik 29 (Peter Lang Verlag, Frankfurt a.M., 1983). · Zbl 0527.90064
[11] H.Th. Jongen, P. Jonker and F. Twilt, ”One-parameter families of optimization problems: equality constraints”,Journal of Optimization Theory and Applications 48 (1986) 141–161. · Zbl 0556.90086
[12] M. Kojima and R. Hirabayashi, ”Continuous deformation of nonlinear programs”,Mathematical Programming Study 21 (1984) 150–198. · Zbl 0569.90074
[13] M. Marcus and H. Minc,A survey of matrix theory and matrix inequalities (Allyn and Bacon Inc., Boston, 1964). · Zbl 0126.02404
[14] J. Milnor,Morse theory, Annals of Mathematics Studies 51 (Princeton University Press, Princeton, NJ, 1963).
[15] D. Siersma, ”Singularities of functions on boundaries, corners etc.”,The Quarterly Journal of Mathematics, Oxford Series (2) 32 (1981) 119–127. · doi:10.1093/qmath/32.1.119
[16] J.E. Spingarn, ”On optimality conditions for structured families of nonlinear programming problems”,Mathematical Programming 22 (1982) 82–92. · Zbl 0478.90067 · doi:10.1007/BF01581027
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