Critical Points Index for Vector Functions and Vector Optimization

In this work, we study the critical points of vector functions from ℝ ⁿ to ℝ ^m 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.