A new algorithm for quadratic integer programming problems with cardinality constraint

摘要

Quadratic integer programming problems with cardinality constraint have many applications in real life. Portfolio selection is an important application in financial optimization. In this paper we develop an exact and efficient algorithm for quadratic integer programming problems with cardinality constraint. This iterative algorithm is actually a branch and bound method, which adopts a domain cut and partition scheme. Removing cardinality constraint and integrality constraints on variables, the relaxation problem is a quadratic programming problem. By solving the quadratic programming problem, we find a lower bound for the optimal objective function value of the primal problem. The domain cut technique is used to cut off some regions that do not contain any feasible solution better than the incumbent solution. Thus the optimality gap is reduced greatly. The finite number of iterations is clear from the fact that there are only a finite number of feasible integer solutions in the feasible region. Encouraging computational results are also reported in the paper.