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. / Graduate

The main purpose of this dissertation is to investigate directional constraint qualifications and necessary optimality conditions for nonsmooth set-constrained mathematical programs. First, we study sufficient conditions for metric subregularity of the set-constrained system. We introduce the directional version of the quasi-/pseudo-normality as a sufficient condition for metric subregularity, which is weaker than the classical quasi-/pseudo-normality, respectively. Then we apply our results to complementarity and Karush-Kuhn-Tucker systems. Secondly, we study directional optimality conditions of bilevel programs. It is well-known that the value function reformulation of bilevel programs provides equivalent single-level optimization problems， which are nonsmooth and never satisfy the usual constraint qualifications such as the Mangasarian-Fromovitz constraint qualification (MFCQ). We show that even the first-order sufficient condition for metric subregularity (which is generally weaker than MFCQ) fails at each feasible point of bilevel programs. We introduce the directional Clarke calmness condition and show that under the directional Clarke calmness condition, the directional necessary optimality condition holds. We perform directional sensitivity analysis of the value function and propose the directional quasi-normality as a sufficient condition for the directional Clarke calmness. / Graduate / 2021-07-07

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