In this thesis, we examine optimization problems with a constraint that allows for only a certain number of variables to be nonzero. This constraint, which is called a cardinality constraint, has received considerable attention in a number of areas such as machine learning, statistics, computational finance, and operations management. Despite their practical needs, most optimization problems with a cardinality constraints are hard to solve due to their nonconvexity. We focus on constructing tight convex relaxations to such problems
