# Parametric Optimization: Singularities, Pathfollowing and by J. Guddat, F. Guerra Vazquez, H. Th. Jongen

(f(x, t) + bT X + txT Ax, H(x, t) + c+ Dx, G(x, t) + e + Fx) belongs to ff**. 3); recall that I={1, ... ,m}, m

5. If J +(x) = J o(x), then x is a non-degenerate critical point (and, hence, stable via the implicit function theorem). e. 3 is violated). The idea behind strong stability relies on the fact that the vanishing of Lagrange parameters Ilj ('linear terms') is compensated by means of ' positive quadratic terms'. l'lL(x) only enlarges its number of positive eigenvalues. This can be seen by means of the following two theorems from linear algebra. Let A be a symmetric n x n matrix and T c ~n a linear subspace.

6) ff** = {(f, H, G)I Each point of LgC belongs to type 1,2,3,4, 5}. 1 (see [118J) The set ff** is C; open and dense in C 3 (lRn x IR, 1R)1 +m+s. Let z denote the general point in IRn x IR, and put z = (x, t), xElR n, tEIR. Define i= 1,2,3,4,5. 2 (see [118J) Let (f,H, G) belong to ff**. c is open and dense in Lgc and, for i = 2, 3,4,5, the set L~c is a discrete point set. We proceed with a description of the five types. For each one we start with conditions that are necessary in order to describe the type under consideration, and we introduce so-called characteristic numbers, which determine the essence of the type.