线性规划单纯形法初探

单纯形法：

将目标函数和约束变换成标准形式。
先找一个基本可行解。
判别它是否为最优解，如果不是，再找一个更好的基本可行解。

例子

min z = x1 + x2
s.t.
2x1 - x2 <= 10 (1)
2 <= x1 <= 20 (2)
0 <= x2 <= 15 (3)

step1: 变换成标准形

==>
(2) ==> 0 <= x1 - 2 <= 20 - 2
取x3 = x1 - 2 ,
则 x1 = x3 + 2 (4)
则 (2) ==> 0 <= x3 <= 18 (5)

(5) ==>
x3 >= 0
x3 <= 18 (6)
(6) ==> x3 + x4 = 18 (7) , x4 >= 0 (8)

(3) ==>
x2 >= 0
x2 <= 15 (9)
(9) ==> x2 + x5 = 15 (10) , x5 >= 0 (11)

将(4)代入(1),得到 2(x3 + 2) - x2 <= 10 (12)
(12) ==> 2x3 + 4 - x2 + x6 = 10 (13) , x6 >= 0

综上，将(4)代入目标函数，消去x1
min z = x3 + 2 + x2 等价于 min z = x3 + x2
约束条件综合起来
(13) ==> -x2 + 2x3 + x6 = 6
(7) x3 + x4 = 18
(10) x2 + x5 = 15
x2 >= 0 , x3 >= 0 , x4 >= 0 , x5 >= 0 , x6 >= 0
以上就是LP的规范形式，即