Abstract
In this work, we study the critical points of vector functions from ℝn to ℝm with n≥m, following the definition introduced by Smale in the context of vector optimization. The local monotonicity properties of a vector function around a critical point which are invariant with respect to local coordinate changes are considered. We propose a classification of critical points through the introduction of a generalized Morse index for a critical point, consisting of a triplet of nonnegative integers. The proposed index is based on the sign of an appropriate invariant vector-valued second-order differential.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnol’D, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps. Birkhauser, Boston (1985)
Smale, S.: Global analysis and economics I: Pareto optimum and a generalization of Morse theory. In: Peixoto, M. (ed.) Dynamical Systems, pp. 531–534. Academic, New York (1973)
Smale, S.: Optimizing several functions. In: Manifolds, Tokyo, 1973, pp. 69–75. University Tokyo Press, Tokyo (1975)
Miglierina, E., Molho, E., Rocca, M.: A Morse-type index for critical points of vector functions. Quaderno di Ricerca 2007/2, Dipartimento di Economia, Universita’ dell’Insubria. http://eco.uninsubria.it/dipeco/Quaderni/files/QF2007_2.pdf
Degiovanni, M., Lucchetti, R., Ribarska, N.: Critical point theory for vector valued functions. J. Convex Anal. 9, 415–428 (2002)
Miglierina, E.: Slow solutions of a differential inclusion and vector optimization. Set-Valued Anal. 12(3), 345–356 (2004)
Jongen, H.Th., Jonker, P., Twilt, F.: Nonlinear Optimization in Finite Dimensions. Morse Theory, Chebyshev Approximation, Transversality, Flows, Parametric Aspects. Kluwer Academic, Dordrecht (2000)
Mather, J.: Stability of C ∞ mappings: VI. The nice dimensions. In: Wall, C.T.C. (ed.) Proc. Liverpool Singularities-Symp. I, pp. 207–253. Springer, Berlin (1971)
Whitney, H.: On singularities of mappings of Euclidean spaces I. Mappings of the plane into the plane. Ann. Math. 62, 374–410 (1955)
Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Academic, Orlando (1985)
Giannessi, F.: Constrained Optimization and Image Space Analysis. Separation of Sets and Optimality Conditions, vol. I. Springer, New York (2005)
Mubarakzyanov, R.G.: Some improvements of the estimates in our paper: Intersection of a space and a polyhedral cone. Sov. Math. (Iz. VUZ) 35(12), 87–89 (1995)
Magnus, J.R., Neudecker, H.: Matrix Differential Calculus with Applications in Statistics and Econometrics. Wiley, Chichester (1999)
Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities. Springer, New York (1973)
Guerraggio, A., Molho, E., Zaffaroni, A.: On the notion of proper efficiency in vector optimization. J. Optim. Theory Appl. 82, 1–21 (1994)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Grundlehren der Mathematischen Wissenschaften, vol. 317. Springer, Berlin (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by T. Rapcsak.
Rights and permissions
About this article
Cite this article
Miglierina, E., Molho, E. & Rocca, M. Critical Points Index for Vector Functions and Vector Optimization. J Optim Theory Appl 138, 479–496 (2008). https://doi.org/10.1007/s10957-008-9383-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-008-9383-5