On the weak stationarity conditions for Mathematical Programs with Cardinality Constraints: a unified approach

TL;DR

This paper investigates stationarity conditions for Mathematical Programs with Cardinality Constraints, proposing a unified approach that encompasses various stationarity notions and addresses both linear and nonlinear cases.

Contribution

It introduces a unified framework for analyzing stationarity conditions in MPCaC, including weaker conditions applicable when standard qualifications fail.

Findings

In the linear case, all minimizers satisfy KKT conditions.

In the nonlinear case, standard constraint qualifications may not hold.

A new hierarchy of stationarity conditions is proposed for MPCaC.

Abstract

In this paper, we study a class of optimization problems, called Mathematical Programs with Cardinality Constraints (MPCaC). This kind of problem is generally difficult to deal with, because it involves a constraint that is not continuous neither convex, but provides sparse solutions. Thereby we reformulate MPCaC in a suitable way, by modeling it as mixed-integer problem and then addressing its continuous counterpart, which will be referred to as relaxed problem. We investigate the relaxed problem by analyzing the classical constraints in two cases: linear and nonlinear. In the linear case, we propose a general approach and present a discussion of the Guignard and Abadie constraint qualifications, proving in this case that every minimizer of the relaxed problem satisfies the Karush-Kuhn-Tucker (KKT) conditions. On the other hand, in the nonlinear case, we show that some standard constraint qualifications may be violated. Therefore, we cannot assert about KKT points. Motivated to find a minimizer for the MPCaC problem, we define new and weaker stationarity conditions, by proposing a unified approach that goes from the weakest to the strongest stationarity.