松弛

形式

$$z=slack(x,y)=\{\begin{array}{ll}max(0,x-y),& \text{计算正松弛}\\ \\ max(0,y-x),& \text{计算负松弛}\\ \\ |x-y|,& \text{计算正负松弛}\end{array}$$

额外变量

$neg\in R-R^{-}$：负松弛。

$pos\in R-R^{-}$：正松弛。

导出符号

$$z=neg+pos$$

数学模型

计算正负松弛

$$\begin{array}{rl}s.t.& x+neg-pos=y\end{array}$$

计算正松弛

$$\begin{array}{rl}s.t.& x-pos\le y\end{array}$$

计算负松弛

$$\begin{array}{rl}s.t.& x+neg\ge y\end{array}$$

样例

kotlin

import kotlinx.coroutines.*
import fuookami.ospf.kotlin.utils.math.*
import fuookami.ospf.kotlin.core.frontend.variable.*
import fuookami.ospf.kotlin.core.frontend.expression.polynomial.*
import fuookami.ospf.kotlin.core.frontend.expression.symbol.linear_function.*
import fuookami.ospf.kotlin.core.frontend.inequality.*
import fuookami.ospf.kotlin.core.frontend.model.mechanism.*
import fuookami.ospf.kotlin.core.backend.plugins.scip.*

val x = RealVar("x")
x.range.leq(Flt64.two)
x.range.geq(-Flt64.three)
val slack = SlackFunction(
    x = x,
    y = Flt64.five,
    name = "slack"
)

val model = LinearMetaModel()
model.add(x)
model.add(slack)
model.minimize(slack)

val solver = ScipLinearSolver()
val result = runBlocking { solver(model) }
assert(result.value!!.obj eq Flt64.three)
assert(result.value!!.solution[0] eq Flt64.two)

完整实现请参考：

• Kotlin

完整样例请参考：

• Kotlin