Cardinality Constrained Optimization Problems

Jinhak Kim, Purdue University

Abstract

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. We first study linear programs with a cardinality constraint (CCLPs). A procedure that yields cutting planes for any given vector that violates the cardinality constraint is developed. These cutting planes are derived from a disjunctive relaxation of the problem. The separation problem is recast as a network optimization problem where the network is constructed from a simplex tableau of the LP relaxation. We then present a procedure to generate a facet-defining inequality of the disjunctive relaxation using a variant of Prim's algorithm. Second, we study an optimization formulation of sparse principal component analysis (sparse PCA). The formulation is a quadratically constrained quadratic problem with a cardinality constraint. The feasible set has a special structure which we call permutation-invariance. This structure allows us to construct the convex hull of the feasible set of the model. The convex hull is written through a majorization inequality that can be modeled using a polynomial number variables and linear inequalities. We then show that sparse PCA can be reformulated as a continuous convex maximization problem without a cardinality constraint. In addition, we derive SDP relaxations for the reformulation. The relaxations are developed based on majorization arguments. The resulting relaxation is provably tighter than the prevalent SDP relaxation proposed by d'Aspremont et al.(2007). Our preliminary computational results show that our SDP relaxation has gaps 90% smaller than those of the classical SDP relaxation. Third, we introduce other approaches for CCLPs. We first present a facial disjunctive reformulation for CCLPs and a finitely-convergent cutting plane algorithm. A generalized reformulation-linearization technique (RLT) is introduced to characterize the convex hull of the feasible set of CCLPs. As a special subclass of CCLP, we study the cardinality-constrainted knapsack problem (CCKP). We developed families of valid inequalities based on disjunctions for the cardinality constraint.

Degree

Ph.D.

Advisors

Tawarmalani, Purdue University.

Subject Area

Management|Operations research

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