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

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.

A sequential optimality condition for Mathematical Programs with Cardinality Constraints

“…In this section we recall some basic definitions and results related to standard nonlinear programming (NLP), as well as the weak stationarity concept proposed in our previous work [12].…”

“…In this paper we propose a sequential optimality condition, associated to the weak stationarity condition presented in our previous work [12], designed to deal with Mathematical Programs with Cardinality Constraints (MPCaC) given by minimize f (x) subject to x ∈ X,…”

“…In [12] we proposed new and weaker stationarity conditions for this class of problems, by means of a unified approach that goes from the weakest to the strongest stationarity. Indeed we cannot assert about KKT points for MPCaC problems, since some standard constraint qualifications are violated.…”

“…. , n. However, the weaker condition proposed in [12], called W I -stationarity, even being weaker than KKT, is not a necessary optimality condition. Therefore, we propose in this work an Approximate Weak stationarity (AW -stationarity) concept, which will be proved to be a legitimate optimality condition, independently of any constraint qualification.…”

“…The paper is organized as follows: in Section 2 we establish the notation, some definitions and results concerning standard nonlinear programming and recall the weak stationarity concept proposed in our previous work [12]. Section 3 presents the main results of this paper, concerning sequential optimality conditions for MPCaC.…”