Cardinality-constrained optimization problems in general position and beyond

TL;DR

This paper investigates the generic properties and structural characteristics of cardinality-constrained optimization problems, establishing conditions for nondegeneracy, stability, and the topological structure of solutions.

Contribution

It demonstrates that CC-LICQ and nondegeneracy are generic properties, describes the global structure of CCOP using Morse theory, and characterizes strong stability through first- and second-order conditions.

Findings

CC-LICQ is generic in CCOP.

Nondegeneracy is a generic property for M-stationary points.

Multiple cells in Morse theory correspond to M-index of stationary points.

Abstract

We study cardinality-constrained optimization problems (CCOP) in general position, i. e. those optimization-related properties that are fulfilled for a dense and open subset of their defining functions. We show that the well-known cardinality-constrained linear independence constraint qualification (CC-LICQ) is generic in this sense. For M-stationary points we define nondegeneracy and show that it is a generic property too. In particular, the sparsity constraint turns out to be active at all minimizers of a generic CCOP. Moreover, we describe the global structure of CCOP in the sense of Morse theory, emphasizing the strength of the generic approach. Here, we prove that multiple cells need to be attached, each of dimension coinciding with the proposed M-index of nondegenerate M-stationary points. Beyond this generic viewpoint, we study singularities of CCOP. For that, the relation between nondegeneracy and strong stability in the sense of Kojima (1980) is examined. We show that nondegeneracy implies the latter, while the reverse implication is in general not true. To fill the gap, we fully characterize the strong stability of M-stationary points under CC-LICQ by first- and second-order information of CCOP defining functions. Finally, we compare nondegeneracy and strong stability of M-stationary points with second-order sufficient conditions recently introduced in the literature.