Global aspects of the continuous reformulation for cardinality-constrained optimization problems

The main goal of this paper is to relate the topologically relevant stationary points of a cardinality-constrained optimization problem and its continuous reformulation up to their type. For that, we focus on the nondegenerate M- and T-stationary points, respectively. Their so-called M- and T-indices, which uniquely determine the global and local structure of optimization problems under consideration in algebraic terms, are traced. As novelty, we suggest to regularize the continuous reformulation for this purpose. The main consequence of our analysis is that the number of saddle points of the regularized continuous reformulation grows exponentially as compared to that of the initial cardinality-constrained optimization problem. Additionally, we obtain the Morse theory for the regularized continuous reformulation by using the corresponding results on mathematical programs with orthogonality type constraints.Comment: 17 page

GAXOTTE, Pierre, Histoire des Français. Collection « L’Histoire », Flammarion, 1951. 2 forts volumes de 580 pages chacun

The feasible set of mathematical programs with complementarity constraints (MPCC) is considered. We discuss local stability of the feasible set with respect to perturbations (up to first order) of the defining functions. Here, stability refers to homeomorphy invariance under small perturbations. For stability we propose a kind of Mangasarian-Fromovitz Condition (MFC) and its stronger version (SMFC). MFC is a natural Constraint Qualification for C-stationarity and SMFC is a generalization of the well-known Clarke's maximal rank condition. It turns out that SMFC implies local stability. MFC and SMFC coincide in case that the number of complementarity constraints (k) equals to the dimension of the state space (n). Moreover, the equivalence of MFC and SMFC is also proven for the cases k=2 as well as under Linear Independence Constraint Qualification (LICQ) for MPCC