论文标题
在某些二次编程问题的对称组上
On Symmetry Groups of Some Quadratic Programming Problems
论文作者
论文摘要
当这些问题在适当的线性转换下对称时，可以简化数学编程问题的解决方案和分析。特别是，对对称性的知识可能有助于减少问题维度，通过线性剪切缩短搜索空间或从先前发现的搜索空间中获得新的局部优点。虽然先前对数学编程中对称性的研究通常涉及解决方案空间坐标的排列，但本文考虑了更大的可逆线性变换。我们研究了二次编程问题的特殊情况，其中目标函数和约束由二次形式给出，并且涉及约束的二次形式的所有矩阵的总和是一个积极的确定矩阵。在这种情况下，仅考虑解决方案空间的正交转换就足够了。在这组正交转换中，我们描述了给出问题对称性的亚组的结构。除此之外，概述了一种找到这种对称性的方法，并在两个简单的示例中进行了说明。
Solution and analysis of mathematical programming problems may be simplified when these problems are symmetric under appropriate linear transformations. In particular, a knowledge of the symmetries may help reduce the problem dimension, cut the search space by linear cuts or obtain new local optima from the ones previously found. While the previous studies of symmetries in the mathematical programming usually dealt with permutations of coordinates of the solutions space, the present paper considers a larger group of invertible linear transformations. We study a special case of the quadratic programming problem, where the objective function and constraints are given by quadratic forms, and the sum of all matrices of quadratic forms, involved in the constraints, is a positive definite matrix. In this setting, it is sufficient to consider only orthogonal transformations of the solution space. In this group of orthogonal transformations, we describe the structure of the subgroup which gives the symmetries of the problem. Besides that, a method for finding such symmetries is outlined, and illustrated in two simple examples.
