最小化受平等和完整性约束的3个变量的总和 [英] Minimizing the sum of 3 variables subject to equality and integrality constraints

问题描述

我正在研究一个编程(使用Python)问题，我必须在3个变量中解决以下类型的线性方程:

I am working on a programming (using Python) problem where I have to solve the following type of linear equation in 3 variables:

x，y，z都是整数.

x, y, z are all integers.

方程式示例:2x + 5y + 8z = 14

条件:Minimize x + y + z

我一直在尝试寻找一种算法，以最佳的方式找到解决方案.如果有人有任何想法，请通过算法或代码源指导我.

I have been trying to search for an algorithm for finding a solution to this, in an optimum way. If anybody has any idea please guide me through algorithm or code-sources.

我很好奇，如果将此问题外推到n个变量，该怎么办?

I am just curious, what can be done if this problem is extrapolated to n variables?

我不想使用hit&试用循环以继续检查值.同样，可能存在方程式无解的情况.

I don't want to use hit & trial loops to keep checking for values. Also, there may be a scenario that equation has no solution.

更新

添加下限条件:

x, y, z >= 0
x, y, z are natural

推荐答案

任何三元组(x，y，z)，其中 z =(14-2x-5y)/8 ，满足您的约束.

Any triple (x, y, z), with z = (14 - 2x - 5y) / 8, satisfies your constraint.

请注意， x + y +(14-2x-5y)/8 从下面是不受限制的.当 x 和 y 均减小时，该函数减小，而没有最小值限制.

Note that x + y + (14 - 2x - 5y) / 8 is unbounded from below. This function decreases when each of x and y decrease, with no finite minimum.

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