ELLIPSOID BOUNDS FOR CONVEX QUADRATIC INTEGER PROGRAMMING

摘要

Solving convex quadratic integer minimization problems by a branch-and-bound algorithm requires tight lower bounds on the optimal objective value. To obtain such dual bounds, we follow the approach of [C. Buchheim, A. Caprara, and A. Lodi, Math. Program., 135 (2012), pp. 369-395] and underestimate the objective function by another convex quadratic function that can be minimized over the integers by simply rounding its continuous minimizer. In geometric terms, we approximate the sublevel sets of the original objective function by auxiliary ellipsoids having the strong rounding property, introduced by [R. Hubner and A. Schobel, European J. Oper. Res., 237 (2014), pp. 404-410]. We first consider axisparallel ellipsoids (corresponding to separable convex quadratic functions) and show how to efficiently compute an axisparallel ellipsoid yielding the tightest lower bound, both in the situation where the location of the continuous minimizer is given and in the situation where it varies, as it happens in a branch-and-bound approach. In the latter case, we compute both worst-case and average-case optimal ellipsoids. Moreover, we consider the class of quasi-round ellipsoids (from Hubner and Schobel). In this case, we do not know whether optimal auxiliary ellipsoids can be computed efficiently. However, we present a heuristic for computing an appropriate quasi-round ellipsoid that still yields valid dual bounds; it turns out that using this approach within a branch-and-bound scheme outperforms the approaches based on axisparallel ellipsoid in terms of running times. We finally combine both approaches by introducing the concept of quasi-axisparallel ellipsoids and present a corresponding extension of the heuristic bound computation.