Disjunctive Optimization: Critical Point Theory

In this paper, we introduce the concepts of (nondegenerate) stationary points and stationary index for disjunctive optimization problems. Two basic theorems from Morse theory, which imply the validity of the (standard) Morse relations, are proved. The first one is a deformation theorem which applies outside the stationary point set. The second one is a cell-attachment theorem which applies at nondegenerate stationary points. The dimension of the cell to be attached equals the stationary index. Here, the stationary index depends on both the restricted Hessian of the Lagrangian and the set of active inequality constraints. In standard optimization problems, the latter contribution vanishes.