On the Solution of Nonconvex Cardinality Boolean Quadratic Programming problems

This paper addresses the solution of a nonlinear boolean quadratic programming problem using three different approaches. The first uses a classic linearization technique to transform the original problem into a Mixed Integer Linear Programming (MILP) problem for which multiple formulations are studied. The second uses the capabilities of current MILP solvers to deal with Mixed Integer Quadratic Programming (MIQP) problems, and the third relies on the utilization of optimization solvers available that can deal with nonlinear combinatorial problems. Two additional strategies relying on the MILP formulations are evaluated. Special emphasis is placed on the definition of computationally efficient MILP reformulations and their comparison with other approaches. The results indicate that the most efficient approach relies on solving the problem as a MIQP problem using a specific MILP solver from the set of solvers tested, while the most efficient MILP formulations are still a better option than solving the problem as a MIQP using the remaining MILP solvers tested, and their performance is considerably superior to the application of general purpose solvers.