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.

翻译：本文研究在椭球体内的混合整数点上最小化一般二次函数的问题。该问题具有强NP难度、难以在常数因子内近似逼近，且最优解可能为无理数。我们的主要结果表明：若固定整数变量个数，可在多项式时间内找到任意精度的高质量解。该算法将Ye、Karmarkar、Vavasis和Zippel证明的信赖域问题多项式可解性自然推广至混合整数场景。这一结果消除了混合整数非线性优化问题中模型信赖域方法设计与分析的关键瓶颈。证明该结果所引入的技术具有独立价值，可应用于其他涉及二次函数的混合整数规划问题。作为应用实例，我们考虑在多面体内的混合整数点上最小化一般二次函数的问题。结果表明：在固定整数变量个数的前提下，可在多项式时间内计算满足弱最优性边界的解。而众所周知，除非P=NP，否则在多项式时间内无法找到满足更强边界的解。