固定维数下的有界三次整数规划

中文摘要 AI 辅助

我们考虑在固定维数下，对有理多胞形的整数点上的有理三次多项式进行精确最小化。Del Pia、Hildebrand、Weismantel 和 Zemmer 证明了在二维情形下该问题可在多项式时间内求解，而四次多项式的最小化在二维情形下已经是 NP 难的。我们证明有界三次情形的可解性可以推广到任意固定维数。证明结合了三次多项式的两个精确恒等式，以及 Ari 和 Hildebrand 最近提出的固定维数整数二次规划算法中的三个要素：对称位移覆盖、二次型的负位移搜索以及整数查询凸可行性。在整数查询点处，负 Hessian 方向产生一个线性曲率割，该割对当前单元中的所有全局最小点均有效；而若不存在这样的方向，则通过精确的端点-Hessian 恒等式产生一个线性目标割。这些割为曲率容许次水平点的凸包定义了一个整数查询分离预言机。通过椭球和格算法实现的整数凸可行性，无需计算格中心点即可获得精确优化。一个实代数局部化界将该结果推广到具有有界实改进次水平集的无界多面体，包括所有强制目标。该论证还解释了为何三次是这种方法的一个自然边界：三次多项式的对称二阶差分恰好是其方向 Hessian，而后者关于基点仿射；对于四次多项式，会出现额外的四阶项。

英文摘要

We give exact polynomial-time algorithms in fixed dimension for mixed-integer quadratic programming over arbitrary rational polyhedra and for integer cubic programming over bounded rational polyhedra. The quadratic objective may be indefinite; the algorithm detects infeasibility, certifies unboundedness, or returns an exact minimizer. For an instance with N variables and m constraints, the running time is 2O(Nlog N) times (m+ 1)O(N) times a factor in which the binary encoding lengths of the coefficients are raised to a power O(N); for pure integer problems, and for mixed-integer problems with integral right-hand side and linear term, only the constraint and quadratic matrices enter this factor, while the right-hand side and linear term enter polynomially. This replaces the dependence on coefficient magnitudes in earlier fixed-parameter results by dependence on encoding length, and the exponent O(N) is optimal under the exponential time hypothesis. For cubic objectives, integer minimization over a bounded polyhedron was previously known to be polynomial only in dimension two. Our algorithm extends to any polynomial whose second directional derivatives are concave in position, which includes cubics with an added concave quartic form. Both algorithms rest on a hidden convexity of such objectives on bounded lattice sets, exposed by a cover of the feasible integer points by reflecting cells. For quadratics, a negative-curvature integer displacement excludes a cell, and otherwise supporting inequalities on its integer points permit exact optimization through an integer-query separation oracle. For cubics and the broader class, negative curvature at a query point yields a linear cut valid for every global minimizer in the cell, and supporting inequalities among the remaining admissible points play the same role.

发表机构

• Grado Department of Industrial and Systems Engineering, Virginia Tech（弗吉尼亚理工大学工业与系统工程系）

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