The index-k-stabilizing differential equation

Abstract

This paper deals with the search for multiple local minima of a differentiable real-valued functionf off variables. Motivated by topological considerations much as Morse Theory — it makes sense to determine critical points¯x forn of index 1 (i.e. exactly one eigenvalue of the HessianD2f(¯x) is negative). For eachk ∈ 0,...,n, the gradient vectorfieldDf ofn is altered — by a partial reflection — into a new vectorfieldFK. Restricted to the critical point set off, only the critical points of indexk are attractors forFK.