求解具有多个变量和不等式约束的多个方程 [英] solving multiple equations with many variables and inequality constraints
问题描述
我正在尝试使用scipy和线性编程来解决许多变量的问题.我有一组变量X，它们是0.5到3之间的实数，我必须解决以下方程式:
I am trying to solve a problem with many variables using scipy and linear programming. I have a set of variables X which are real numbers between 0.5 and 3 and I have to solve the following equations :
346 <= x0*C0 + x1*C1 + x2*C2 +......xN*CN <= 468
25 <= x0*p0 + x1*p1 + x2*p2 +......xN*pN <= 33
12 <= x0*c0 + x1*c1 + x2*c2 +......xN*cN <= 17
22 <= x0*f0 + x1*f1 + x2*f2 +......xN*fN <= 30
数字C0 ... CN，p0 ... pN，c0 ... cN，f0 ... fN已经给我了.我尝试通过以下方式解决此问题:
the numbers C0...CN , p0...pN , c0...cN , f0...fN are already given to me. I tried to solve this in the following way:
import numpy as np
from scipy.optimize import linprog
from numpy.linalg import solve
A_ub = np.array([
[34, 56, 32, 21, 24, 16, 19, 22, 30, 27, 40, 33],
[2, 3, 2, 1.5, 3, 4, 1, 2, 2.5, 1, 1.2, 1.3],
[1, 2, 3, 1.2, 2, 3, 0.6, 1, 1, 1.2, 1.1, 0.8],
[0.5, 2, 2, 1, 3, 4, 1, 1, 1, 0.5, 0.3, 1.2],
[-34, -56, -32, -21, -24, -16, -19, -22, -30, -27, -40, -33],
[-2, -3, -2, -1.5, -3, -4, -1, -2, -2.5, -1, -1.2, -1.3],
[-1, -2, -3, -1.2, -2, -3, -0.6, -1, -1, -1.2, -1.1, -0.8],
[-0.5, -2, -2, -1, -3, -4, -1, -1, -1, -0.5, -0.3, -1.2]])
b_ub = np.array([468, 33, 17, 30, -346, -25, -12, -22])
c = np.array([34, 56, 32, 21, 24, 16, 19, 22, 30, 27, 40, 33])
res = linprog(c, A_eq=None, b_eq=None, A_ub=A_ub, b_ub=b_ub, bounds=(0.5, 3))
方程式的说明A_ub的第一行与b_ub相同，因为我们正在尝试使方程式最大化，并确保其在给定的边界范围内，即468和346，这意味着我想获取该值尽可能接近上限.
Explanation for the equations the first row of A_ub is the same as b_ub, as we are trying to maximize the equation as well as make sure it is within the given boundary limits i.e 468 and 346 meaning that I want to get the value as close as possible to the upper limit.
我将[-34, -56, -32, -21, -24, -16, -19, -22, -30, -27, -40, -33]放在A_ub中，将-346放在b_ub中，其逻辑是:
I put [-34, -56, -32, -21, -24, -16, -19, -22, -30, -27, -40, -33] in A_ub and -346 in b_ub with the logic :
-346 > -(x0*C0 + x1*C1 + x2*C2 +......xN*CN)将解决该方程的下界问题.我对其余的人也一样.
-346 > -(x0*C0 + x1*C1 + x2*C2 +......xN*CN) which would solve the problem of lower bounds for the equation. I do the same with the rest of them.
但是我觉得我的方法是错误的，因为我得到的答案是0.425，而nan是res.x
But I feel my approach is wrong as I get the answer as 0.425 for res.fun and nan as the value of res.x
x的上限为3，下限为0.5
The upper bound for x is 3 and the lower bound is 0.5
我如何如上所述定义问题，以便在牢记上限的情况下获得接近468的最大值?如何使用scipy定义下限?我是第一次进行线性编程，所以我可能错过了可以帮助我的想法.
How do I define the problem as shown above in order to get a maximum value close to 468 while keeping in mind the upper bounds? How do I define lower bounds using scipy? I am working on linear programming for the first time so I may have missed out on ideas that can help me out.
我也愿意接受其他任何解决方案.
I am also open to any other solutions.
推荐答案
这种不平等系统是不可行的:没有一种解决方案可以满足所有约束条件.您可以从res看到这一点:
This system of inequalities is not feasible: there is no solution that satisfies all constraints. You can see that from res:
fun: 0.42500000000000243
message: 'Optimization failed. Unable to find a feasible starting point.'
nit: 28
status: 2
success: False
x: nan
我相信这是正确的结果(我已使用另一个LP系统对此进行了验证).
I believe this is a correct result (I verified this with another LP system).
注意:如果将边界更改为(0,3)，则将获得可行的解决方案.
Note: if you change the bounds to (0,3), you will get a feasible solution.
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