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. As a result, our findings pave the way for designing efficient trust region methods for mixed integer nonlinear optimization problems. The techniques that we introduce are of independent interest and can be used in other mixed integer nonlinear optimization problems. 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.