If Function

Function Form

$$y=\text{If}(x\cdot rhs)=\{\begin{array}{ll}1,& x\cdot rhs\\ \\ 0,& \mathrm{\neg }(x\cdot rhs)\end{array}$$

where $\cdot$ can be $\le$, $\ge$, or $=$.

Additional Variables

$k_i\in [0,1]$: Linear piecewise weights.

$y^{\prime }\in \{0,1\}$: Logical value of the inequality.

Derived Symbol

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

Mathematical Model

If $\cdot$ is $\le$, we have:

$$\text{s.t.}\{\begin{array}{ll}\{\begin{array}{l}x=k_0\cdot min(x)+k_1\cdot rhs+k_2\cdot max(x)\\ \\ k_0\le y^{\prime }\\ \\ k_2\le 1-y^{\prime }\\ \\ k_2\ge ϵ\cdot (1-y^{\prime })\end{array},& rhs\in [min(x),max(x)]\\ \\ y^{\prime }=\text{bin}(max(x)\le rhs),& rhs\notin [min(x),max(x)]\end{array}$$

If $\cdot$ is $\ge$, we have:

$$\text{s.t.}\{\begin{array}{ll}\{\begin{array}{l}x=k_0\cdot min(x)+k_1\cdot rhs+k_2\cdot max(x)\\ \\ k_0\le 1-y^{\prime }\\ \\ k_0\ge ϵ\cdot (1-y^{\prime })\\ \\ k_2\ge y^{\prime }\end{array},& rhs\in [min(x),max(x)]\\ \\ y^{\prime }=\text{bin}(max(x)\le rhs),& rhs\notin [min(x),max(x)]\end{array}$$

If $\cdot$ is $=$, we have:

$$\text{s.t.}\{\begin{array}{ll}\{\begin{array}{l}x=k_0\cdot min(x)+k_1\cdot rhs+k_2\cdot max(x)\\ \\ k_0+k_2\le 1-y^{\prime }\\ \\ k_0+k_2\ge ϵ\cdot (1-y^{\prime })\end{array},& rhs\in [min(x),max(x)]\\ \\ y^{\prime }=\text{bin}(max(x)\le rhs),& rhs\notin [min(x),max(x)]\end{array}$$

where,

$$\text{bin}(cond)=\{\begin{array}{l}1,cond\\ \\ 0,\mathrm{\neg }cond\end{array}$$

and $ϵ$ satisfies the following property:

$$(ϵ=0)\iff ((x=rhs)\Rightarrow \mathrm{\neg }(x\cdot rhs))$$

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.geq(Flt64.two)
x.range.leq(Flt64.five)
val condition1 = IfFunction(x geq Flt64.three, name = "c1")
val condition2 = IfFunction(x geq Flt64.three, epsilon = Flt64.zero, name = "c2")
val condition3 = IfFunction(x leq Flt64.one, name = "c3")
val solver = ScipLinearSolver()

val model1 = LinearMetaModel()
model1.add(x)
model1.add(condition1)
model1.addConstraint(condition1 eq true)
model1.minimize(x)
val result1 = runBlocking { solver(model1) }
assert(result1.value!!.obj eq Flt64.three)

val model2 = LinearMetaModel()
model2.add(x)
model2.add(condition1)
model2.addConstraint(condition1 eq false)
model2.maximize(x)
val result2 = runBlocking { solver(model2) }
assert(result2.value!!.obj ls Flt64.three)

val model3 = LinearMetaModel()
model3.add(x)
model3.add(condition2)
model3.addConstraint(condition2 eq false)
model3.maximize(x)
val result3 = runBlocking { solver(model3) }
assert(result3.value!!.obj eq Flt64.three)

val model4 = LinearMetaModel()
model4.add(x)
model4.add(condition3)
model4.maximize(condition3)
val result4 = runBlocking { solver(model4) }
assert(result4.value!!.obj eq Flt64.zero)

Complete Implementation Reference:

• Kotlin

Complete Example Reference:

• Kotlin