The Mixed Integer Trust Region Problem

In this paper we consider the problem of minimizing a general quadratic function over the mixed integer points in an ellipsoid. This problem is strongly NP-hard, NP-hard to approximate within a constant factor, and optimal solutions can be irrational. In our main result we show that an arbitrarily good solution can be found in polynomial time, if we fix the number of integer variables. This algorithm provides a natural extension to the mixed integer setting, of the polynomial solvability of the trust region problem proven by Ye, Karmarkar, Vavasis, and Zippel. Our result removes a key bottleneck in the design and analysis of model trust region methods for mixed integer nonlinear optimization problems. The techniques introduced to prove this result are of independent interest and can be used in other mixed integer programming problems involving quadratic functions. As an example we consider the problem of minimizing a general quadratic function over the mixed integer points in a polyhedron. For this problem, we show that a solution satisfying weak bounds with respect to optimality can be computed in polynomial time, provided that the number of integer variables is fixed. It is well-known that finding a solution satisfying stronger bounds cannot be done in polynomial time, unless P=NP.