Improving the performance of MIQP solvers for quadratic programs with cardinality and minimum threshold constraints: A semidefinite program approach

We consider in this paper quadratic programming problems with cardinality and minimum threshold constraints that arise naturally in various real-world applications such as portfolio selection and subset selection in regression. This class of problems can be formulated as mixed-integer 0-1 quadratic programs. We propose a new semidefinite program (SDP) approach for computing the "best" diagonal decomposition that gives the tightest continuous relaxation of the perspective reformulation of the problem. We also give an alternative way of deriving the perspective reformulation by applying a special Lagrangian decomposition scheme to the diagonal decomposition of the problem. This derivation can be viewed as a "dual" method to the convexification method employing the perspective function on semicontinuous variables. Computational results show that the proposed SDP approach can be advantageous for improving the performance of mixed-integer quadratic programming solvers when applied to the perspective reformulations of the problem.