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 non-degenerate M- and T-stationary points, respectively. Their M-index and T-index, respectively, 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. This growth appears to be inherent and is reproduced in other relaxation attempts.