Combining Discrete Ellipsoid-based Search and Branch-and-Cut for Integer Least Squares Problems

In this paper we perform computational studies on two types of integer least squares (ILS) problems that arise in wireless communication applications: unconstrained and box-constrained ILS problems. We perform extensive experiments comparing the performance of the discrete ellipsoid-based search (DEBS) method and the branch-and-cut (B&C) mixed integer programming (MIP) solver, and identify the key characteristics that affect the performance of each approach. We then extend a hybrid algorithm originally designed for the binary quadratic programming (BQP) problem to solve the two types of ILS problems. The hybrid algorithm integrates techniques from the DEBS method and the B&C based MIP solver. Our numerical results demonstrate that MIP and DEBS perform quite differently on different types of problems. The DEBS method performs consistently more efficient than both the state-of-the-art MIP solvers for the unconstrained problems. However the MIP solvers are superior on some box-constrained problems with large noise in the data. Results also show that the new hybrid algorithm is promising: achieving the same or better performance than both the DEBS method and one MIP solver while under-performing the other MIP solver on the box-constrained problems with certain properties. W.-Y. Ku and J. C. Beck are with the Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON, M5S3G8 CANADA e-mail: [email protected], [email protected]