Semi-Function (Positive Part)

Function Form

$$y=\text{semi}(x)=max(0,x)=\{\begin{array}{ll}x,& x>0\\ \\ 0,& x\le 0\end{array}$$

Constants

$$m=max(|x|)$$

Additional Variables

$u\in \{0,1\}$: Indicator for $x>0$.

$y^{\prime }\in \mathbb{R}-{\mathbb{R}}^{-}$: Represents $max(0,x)$.

Derived Symbol

$$y=y^{\prime }$$

Mathematical Model

$$\begin{array}{rl}\text{s.t.}& y\ge x\\ \\ & y\le x+m\cdot u\\ \\ & y\le m\cdot (1-u)\end{array}$$

Code Example

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 = URealVar("x")
x.range.leq(Flt64.three)
val y = URealVar("y")
y.range.geq(Flt64.two)
y.range.leq(Flt64.five)
val semi = SemiFunction(
    x - y,
    name = "semi"
)
val solver = ScipLinearSolver()

val model1 = LinearMetaModel()
model1.add(x)
model1.add(y)
model1.add(semi)
model1.minimize(semi)

val result1 = runBlocking { solver(model1) }
assert(result1.value!!.obj eq Flt64.zero)
assert(result1.value!!.solution[1] geq result1.value!!.solution[0])

val model2 = LinearMetaModel()
model2.add(x)
model2.add(y)
model2.add(semi)
model2.maximize(semi)

val result2 = runBlocking { solver(model2) }
assert(result2.value!!.obj eq Flt64.one)
assert(result2.value!!.solution[0] eq Flt64.three)
assert(result2.value!!.solution[1] eq Flt64.two)

Complete Implementation Reference:

• Kotlin

Complete Example Reference:

• Kotlin