2019.11.30-组合优化问题的数学建模-线性化非线性表达式1
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Converting Non-linear Models into Linear Ones
“If-then” Logic (Implication)
Variable
𝑥,𝑧
∈
[
𝑎,𝑏
]
,𝑦
∈
{
0,1
}
Constant
𝑎,𝑏
∈
𝑅
𝑀
>
max
(
|
𝑎
|
,
|
𝑏
|
,𝑏
―
𝑎
)
Upper bound
Lower bound
Usage
𝑧
≤
𝑥
⋅
𝑦⇒
𝑧
≤
𝑥
+
𝑀
⋅
(1
―
𝑦)
𝑧
≤
𝑀
⋅
𝑦
𝑧
≥
𝑥
⋅
𝑦⇒
𝑧
≥
𝑥
―
𝑀
⋅
(1
―
𝑦)
𝑧
≥
―
𝑀
⋅
𝑦
Reference
http://math.stackexchange.com/questions/112159/what-are-the-
algorithms-for-integer-programming-in-which-constraints-are-
depende/112927#112927
Boolean Logic on Variables
Variable
𝑥,𝑦
∈
{
0,1
}
Constant
Upper bound
Lower bound
Usage
𝑦
=
𝑥
1
∧
𝑥
2
∧
…
∧
𝑥
𝑛
⇒0
≤
𝑛
𝑖
=
0
𝑥
𝑖
―
𝑛
⋅
𝑦
≤
𝑛
―
1
𝑦
=
𝑥
1
∨
𝑥
2
∨
…
∨
𝑥
𝑛
⇒0
≤
𝑛
⋅
𝑦
―
𝑛
𝑖
=
0
𝑥
𝑖
≤
𝑛
―
1
Reference
https://cs.stackexchange.com/questions/12102/express-boolean-logic-
operations-in-zero-one-integer-linear-programming-ilp
Boolean Logic on Constraints
Variable
𝑥
𝑖𝑗
∈
𝑅,
𝑧
𝑖
∈
{
0,1
}
Constant
𝑎
𝑖𝑗
,
𝑏
𝑖
∈
𝑅
𝑀
≥
max
i
𝑗
𝑎
𝑖𝑗
⋅
𝑥
𝑖𝑗
―
𝑏
𝑖
Upper bound
Lower bound
Usage
𝑖
𝑗
𝑎
𝑖𝑗
⋅
𝑥
𝑖𝑗
≤
𝑏
𝑖
⇒
𝑗
𝑎
𝑖𝑗
⋅
𝑥
𝑖𝑗
≤
𝑏
𝑖
+
𝑀
⋅
(
1
―
𝑧
𝑖
)
, ∀𝑖
𝑖
𝑧
𝑖
≥
1
Reference
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