Optimality Conditions for Cardinality Constrained Optimization Problems
Date
2022-08-11
Authors
Xiao, Zhuoyu
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Abstract
Cardinality constrained optimization problems (CCOP) are a new class of optimization
problems with many applications. In this thesis, we propose a framework
called mathematical programs with disjunctive subspaces constraints (MPDSC), a
special case of mathematical programs with disjunctive constraints (MPDC), to investigate
CCOP. Our method is different from the relaxed complementarity-type reformulation
in the literature. The first contribution of this thesis is that we study various stationarity conditions for MPDSC, and then apply them to CCOP. In particular, we recover disjunctive-type strong (S-) stationarity and Mordukhovich (M-) stationarity for CCOP, and then reveal the relationship between them and those from the relaxed complementarity-type reformulation. The second contribution of this thesis is that we obtain some new results for MPDSC, which do not hold for MPDC in general. We show that many constraint qualifications like the relaxed constant positive linear dependence (RCPLD) coincide with their piecewise versions for MPDSC. Based on such result, we prove that RCPLD implies error bounds for MPDSC. These two results also hold for CCOP. All of these disjunctive-type constraint qualifications for CCOP derived from MPDSC are weaker than those from the relaxed complementarity-type reformulation in some sense.
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Keywords
cardinality constrained optimization problems, disjunctive subspaces constraints, necessary optimality conditions, constraint qualifications, metric subregularity, error bounds property