Improving the Performance of MIQP Solvers for Quadratic Programs with Cardinality and Minimum Threshold Constraints: A Semidefinite Program Approach

Abstract

We consider in this paper quadratic programming problems with cardinality and minimum threshold constraints which 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 semi-continuous variables. Computational results show that the proposed SDP approach can be advantageous for improving the performance of MIQP solvers when applied to the perspective reformulations of the problem.

Improving the Approximated Projected Perspective Reformulation by dual information

“…Since (MV) is a non-separable MIQP, a diagonal matrix D has to be determined such that Q − D is positive semidefinite: the PR technique is applied to i∈N D ii x 2 i , leaving the remaining part x T (Q − D)x untouched. Choosing D is nontrivial: one can use e.g., a "small" SDP as advocated in [5], or a "large" SDP as proposed in [10]. [10].…”

“…Choosing D is nontrivial: one can use e.g., a "small" SDP as advocated in [5], or a "large" SDP as proposed in [10]. [10].…”

“…For our tests we used the 90 randomly-generated instances, 30 for each value of n ∈ {200, 300, 400}, already employed in [1,4,5,6,10] to which the interested reader is referred for details. Here we only remark that " + " instances are strongly diagonally dominant, " 0 " ones are weakly diagonally dominant, and " − " ones are not diagonally dominant; the less diagonally dominant, the harder an instance is.…”

“…Here we only remark that " + " instances are strongly diagonally dominant, " 0 " ones are weakly diagonally dominant, and " − " ones are not diagonally dominant; the less diagonally dominant, the harder an instance is. We have set k = 10, as in [1,10]; this is a "tight" value, since the maximum number of assets that the model can choose, due to the lower limits x i > 0, without the cardinality constraint is ≈ 20 for all n. The (MV) instances and the diagonals used in the experiments are available at http://www.di.unipi.it/optimize/Data/MV.html.…”

On Valid Inequalities for Quadratic Programming with Continuous Variables and Binary Indicators

“…Specifically, Anstreicher [1] shows that using a subset of all valid inequalities for Q suffices to solve 49 of 50 instances (up to size n = 60) of the BoxQP library [12] at the root node. The inequalities used in the study of Anstreicher are all concave-QPB inequalities and posdiag-QPB inequalities derived via the Reformulation-Linearization Technique [26] and the triangle inequalities for BQP introduced by [23].…”