# Critical sets in parametric optimization

DOI: 10.1007/BF01582234

- Cite this article as:
- Jongen, H.T., Jonker, P. & Twilt, F. Mathematical Programming (1986) 34: 333. doi:10.1007/BF01582234

## Abstract

We deal with one-parameter families of optimization problems in finite dimensions. The constraints are both of equality and inequality type. The concept of a ‘generalized critical point’ (g.c. point) is introduced. In particular, every local minimum, Kuhn-Tucker point, and point of Fritz John type is a g.c. point. Under fairly weak (even generic) conditions we study the set*∑* consisting of all g.c. points. Due to the parameter, the set*∑* is pieced together from one-dimensional manifolds. The points of*∑* can be divided into five (characteristic) types. The subset of ‘nondegenerate critical points’ (first type) is open and dense in*∑* (nondegenerate means: strict complementarity, nondegeneracy of the corresponding quadratic form and linear independence of the gradients of binding constraints). A nondegenerate critical point is completely characterized by means of four indices. The change of these indices along*∑* is presented. Finally, the Kuhn-Tucker subset of*∑* is studied in more detail, in particular in connection with the (failure of the) Mangasarian-Fromowitz constraint qualification.