A Morse-type index for critical points of vector functions
AbstractIn this work we study the critical points of vector functions form Rn to Rm with n m, following the definition introduced by S. 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 an index for a critical point consisting of a triple of nonnegative integers. The proposed index is based on the ”sign” of an appropriate vector-valued second-order differential, that is proved to be invariant with respect to local coordinate changes. In order to avoid anomalous behaviours of the Jacobian matrix, the analysis is partially restricted to the proper critical points, a subset of critical points which enjoy stability properties with respect to perturbations of the order structure. Under nondegeneracy conditions, the index is proved to be locally constant. Moreover, the stability properties of the index with respect to perturbations both of the ordering cone and of the function are considered. Finally, the consistency of the proposed classification with the one given by Whitney for stable maps from the plane into the plane is proved. Keywords: Copula; Fréchet class; positive dependence stochastic ordering; right-tail decreasing (RTI); left-tail decreasing (LTD)
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Bibliographic InfoPaper provided by Department of Economics, University of Insubria in its series Economics and Quantitative Methods with number qf0702.
Length: 25 pages
Date of creation: Feb 2007
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This paper has been announced in the following NEP Reports:
- NEP-ALL-2007-04-28 (All new papers)
- NEP-CSE-2007-04-28 (Economics of Strategic Management)
- NEP-EEC-2007-04-28 (European Economics)
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