Absolute Value

Function Form

$$y=\text{Abs}(x)=|x|$$

Constant Definition

$$M=max(|x|)$$

Additional Variables

$neg\in [0,1]$: indicates the relative distance to the lower bound in the negative direction.

$pos\in [0,1]$: indicates the relative distance to the upper bound in the positive direction.

$p\in \{0,1\}$: sign indicator variable.

Basic Formula

$$y=M\cdot pos+M\cdot neg$$

Mathematical Model

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

The above model only provides an upper bound for $y$ relative to $|x|$, suitable for minimization objective functions. If precise $y$ values are required in equality constraints or maximization objective functions, the following mathematical model must be additionally appended:

$$\begin{array}{rlrl}\text{s.t.}& neg+pos& \le & 1\\ & pos& \le & p\\ & neg& \le & 1-p\end{array}$$

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 = RealVar("x")
val abs = AbsFunction(x, name = "abs")

x.range.leq(Flt64.two)
x.range.geq(-Flt64.three)

val model = LinearMetaModel()
model.add(x)
model.add(abs)
model.maximize(abs)

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

Complete Implementation Reference:

• Kotlin

Complete Example Reference:

• Kotlin