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

TL;DR

This paper explores the relationship between stationary points of cardinality-constrained optimization problems and their continuous reformulations, introducing regularization to analyze their topological structure and saddle point growth.

Contribution

It introduces a regularization approach to relate stationary points of original and reformulated problems, and derives Morse theory for the regularized reformulation.

Findings

Number of saddle points grows exponentially after regularization

Regularization helps relate stationary points between original and reformulated problems

Morse theory is established for the regularized reformulation

Abstract

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.