论文标题
封闭凸层的谐波中心：定义，计算和某些属性
Harmonic Center of a Closed Convex Polytope: Definition, Calculation and Some Properties
论文作者
论文摘要
考虑了由M线性不等式约束定义的n维中的封闭凸多属。如果L是从任何可行点P沿任何方向绘制的直线，则通常会在一个点相交，从而提供M相交。结果表明，沿着这条线的某个地方存在一个唯一的可行点q，因此1/di值的总和为0，其中di是q和与约束i的交点之间的代数距离，沿线测量。点Q定义为沿线L的谐波点。多层室的谐波中心定义为那个点，这是所有通过其绘制的N线的谐波点，每个线平行于坐标轴之一。显示了这种中心的存在和独特性。可以使用坐标搜索算法（CS）来计算谐波中心，如一些简单的示例所示。此处定义的谐波中心是对前面定义的BI中心的概括，并且在几个方面都更好。结果表明，多层的谐波中心也是任何方向绘制的任何线的谐波点。还表明，对于任何严格可行的点P，都存在一个独特的谐波超平面穿过它，因此P是任何位于谐波超平面并通过P的线的谐波点。
A closed convex polytope in n dimensions defined by m linear inequality constraints is considered. If L is a straight line drawn in any direction from any feasible point P, then in general, it intersects every constraint at one point, giving m intersections. It is shown that there exists a unique feasible point Q somewhere along this line, such that the sum of 1/di values is 0, where di is the algebraic distance between Q and the intersection with constraint i, measured along the line. The point Q is defined as the harmonic point along the line L. The harmonic center of the polytope is defined as that point which is the harmonic point for all n lines drawn through it, each parallel to one of the coordinate axes. The existence and uniqueness of such a center is shown. The harmonic center can be calculated using the coordinate search algorithm (CS), as illustrated with some simple examples. The harmonic center defined here is a generalization of the BI center defined earlier and is better in several respects. It is shown that the harmonic center of the polytope is also the harmonic point for any line drawn through it in any direction. It is also shown that for any strictly feasible point P, there exists a unique harmonic hyperplane passing through it, such that P is the harmonic point for any line which lies in the harmonic hyperplane and passes through P.
