An Exact Algorithm for Nonconvex Quadratic Integer Minimization Using Ellipsoidal Relaxations

Abstract

We propose a branch-and-bound algorithm for minimizing a not necessarily convex quadratic function over integer variables. The algorithm is based on lower bounds computed as continuous minima of the objective function over appropriate ellipsoids. In the nonconvex case, we use ellipsoids enclosing the feasible region of the problem. In spite of the nonconvexity, these minima can be computed quickly; the corresponding optimization problems are equivalent to trust-region subproblems. We present several ideas that allow to accelerate the solution of the continuous relaxation within a branch-and-bound scheme and examine the performance of the overall algorithm by computational experiments. Good computational performance is shown especially for ternary instances.

A fast branch-and-bound algorithm for non-convex quadratic integer optimization subject to linear constraints using ellipsoidal relaxations

“…we assumed x 0 = 0. We show that Problem (2) can be transformed into a trust-region type problem so that the branch-and-bound algorithm defined in [2] can be applied. Let us write vectors and matrices accordingly to the partition induced by B and N. Thus, the x B variables can be eliminated via substituting…”

“…This approach can be embedded into the branch-and-bound procedure proposed in [2], where the enumeration strategy is depth-first and branching is done by fixing the value of the variables in a predetermined order. By the latter restriction, we ensure that the matrices Q and H only depend on the depth of the node in the branch-and-bound tree, i.e., on which variables have been fixed so far, but not on their values: first, a basis B 0 of A can be computed in the preprocessing phase.…”

“…This implies that only n different matrices Q and H appear in the entire branch-and-bound tree, so that, similarly to [2], all time-consuming calculations concerning Q and H can be performed in a preprocessing phase.…”

“…For the variant of (1) containing only box constraints, but no other linear constraints, Buchheim et al [2] recently proposed a branch-and-bound algorithm based on ellipsoidal relaxations of the feasible box. More precisely, a suitable ellipsoid E containing [l, u] is determined and q(x) is minimized over x ∈ E.…”

“…A finite and computable real valueM > 0 exists such that the resulting quadratic problem with only the simple constraints [l, u] ∩ Z n is equivalent to (1). Thus the branch-and-bound algorithm defined in [2] which uses an ellipsoidal relaxation E of the feasible set can be used in a straighforward way.…”