On the Critical Sets of One-Parameter Quadratic Optimization Problems

Abstract

We consider the optimization of a quadratic objective function under linear constraints, where all appearing coefficients are continuously differentiable functions of one real variable (the parameter). A generalized critical point (g.c.point) is a feasible point for which the gradient of the object function together with the gradients of the active constraints form a linear dependent set of vectors. We prove that under weak (even generic) conditions the g.c.points classify into three different types. Points of the first type are just the non-degenerate critical points, where “non-degeneracy” means: linear independence of the active constraints (LICQ), strict cornplementarity (ND 1), and non-degeneracy of the quadratic form (on the linearized feasible set) associated with the Hessian of the Lagrange function (ND 2) In the cases of the other two types, either LICQ and ND 2 hold but ND 1 not, or LICQ is violated and the total number of active constraints equals n + 1.This paper is closely related to results obtained by Jongen, Jonker and Twilt concerning more general one-parameter smooth optimization problems in ℝ n with finitely many (in-)equality constraints. In fact, when restricting ourselves to quadratic problems, two of the five types as introduced by Jongen et al are ruled out, whereas no “new” types do appear.Keywords(generalized) critical pointMorse indicesquadratic optimizationsensitivity analysis