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Mathematical Programs with Cardinality Constraints

Ademir Alves Ribeiro (UFPR)

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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. We reformulate MPCaC in a suitable way, by modeling it as a nonlinear programming problem, which will be referred to as relaxed problem. Since the standard constraint qualifications may be violated, we cannot assert about KKT points. Motivated to find a minimizer for the MPCaC problem, we then define new and weaker stationarity conditions. We also propose a comparative study of sequential optimality conditions for Mathematical Programs with Cardinality Constraints. Besides analyzing some of the classical approximate conditions for nonlinear programming, such as AKKT, CAKKT and PAKKT, we also propose an Approximate Weak stationarity (AW-stationarity) concept designed to deal with this class of problems and we prove that it is a legitimate optimality condition independently of any constraint qualification. Finally, we define two tailored (strong and weak) second-order necessary conditions, MPCaC-SSONC and MPCaC-WSONC and propose a constraint qualification (CQ), namely, MPCaC-relaxed constant rank constraint qualification (MPCaC-RCRCQ), and establish the validity of MPCaC-SSONC at minimizers under this new CQ. Moreover, we discuss the application of a second-order augmented Lagrangian algorithm on MPCaCs and prove its global convergence to MPCaC-WSONC points under our MPCaC-RCRCQ condition.

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