Univariate Linear Piecewise Function

Function Form

$$y=\text{Ulp}(x)=k_{i}x+b_i,x\in [a_i,b_i],i=0,1,2,\dots$$

Constants

Let the point set of this linear piecewise function be $P$, where each point $i$ corresponds to the value $(x_i,y_i)$.

Additional Variables

$k_i\in [0,1]$: Relative distance to point $i$.

$b_i\in \{0,1\}$: Indicator for being on the line segment formed by points $i$ and $i+1$.

Derived Symbol

$$y=\sum _{i\in P}{k}_i\cdot y_i$$

Mathematical Model

$$\begin{array}{rlrl}\text{s.t.}& x& =& \sum _{i\in P}{k}_i\cdot x_i\\ & \sum _{i\in P}{k}_i& =& 1\\ & \sum _{i\in P}{b}_i& =& 1\\ & k_i& \le & b_{i-1}+b_i,& \mathrm{\forall }i\in P\end{array}$$

Code Example

kotlin

import kotlinx.coroutines.*
import fuookami.ospf.kotlin.utils.math.*
import fuookami.ospf.kotlin.utils.math.geometry.*
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.model.mechanism.*
import fuookami.ospf.kotlin.core.backend.plugins.scip.*

val x = URealVar("x")
x.range.leq(Flt64.two)

val ulp = UnivariateLinearPiecewiseFunction(
    x = x,
    points = listOf(
        point2(),
        point2(x = Flt64.one, y = Flt64.two),
        point2(x = Flt64.two, y = Flt64.one)
    ),
    name = "y"
)

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

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

Complete Implementation Reference:

• Kotlin

Complete Example Reference:

• Kotlin