In pure integer linear programming it is often desirable to work with polyhedra that are full-dimensional, and it is well known that it is possible to reduce any polyhedron to a full-dimensional one in polynomial time. More precisely, using the Hermite normal form, it is possible to map a non full-dimensional polyhedron to a full-dimensional isomorphic one in a lower-dimensional space, while preserving integer vectors. In this paper, we extend the above result simultaneously in two directions. First, we consider mixed integer vectors instead of integer vectors, by leveraging on the concept of "integer reflexive generalized inverse." Second, we replace polyhedra with convex quadratic sets, which are sets obtained from polyhedra by enforcing one additional convex quadratic inequality. We study structural properties of convex quadratic sets, and utilize them to obtain polynomial time algorithms to recognize full-dimensional convex quadratic sets, and to find an affine function that maps a non full-dimensional convex quadratic set to a full-dimensional isomorphic one in a lower-dimensional space, while preserving mixed integer vectors. We showcase the applicability and the potential impact of these results by showing that they can be used to prove that mixed integer convex quadratic programming is fixed parameter tractable with parameter the number of integer variables. Our algorithm unifies and extends the known polynomial time solvability of pure integer convex quadratic programming in fixed dimension and of convex quadratic programming.

翻译：在纯整数线性规划中，通常希望处理全维多面体，且众所周知，可以在多项式时间内将任意多面体约化为全维多面体。具体而言，利用埃尔米特标准型，可将非全维多面体映射到低维空间中的全维同构多面体，同时保持整数向量不变。本文从两个方向同时推广上述结果。首先，我们通过引入"整数反射广义逆"的概念，将整数向量推广为混合整数向量。其次，我们将多面体替换为凸二次集合——即通过添加一个凸二次不等式约束从多面体导出的集合。我们研究了凸二次集合的结构性质，并利用这些性质设计了多项式时间算法：用于识别全维凸二次集合，以及寻找一个仿射函数，将非全维凸二次集合映射到低维空间中的全维同构集合，同时保持混合整数向量不变。通过证明这些结果可用于论证混合整数凸二次规划是关于整数变量个数的固定参数可解问题，我们展示了其适用性与潜在影响。该算法统一并扩展了已知的固定维数纯整数凸二次规划与凸二次规划的多项式时间可解性。