平衡三值化

形式

$$y=BTer(x)=\{\begin{array}{ll}1,& x\\ \\ -1,& \mathrm{\neg }x\\ \\ 0,& \text{otherwise}\end{array}$$

离散

常量

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

额外变量

$y_p^{\prime }\in \{0,1\}$ ：多项式为真的逻辑值。

$y_n^{\prime }\in \{0,1\}$ ：多项式为假的逻辑值。

导出符号

$$y=y_p^{\prime }-y_n^{\prime }$$

数学模型

$$\begin{array}{rlrl}s.t.& m\cdot y_p^{\prime }& \ge & x\\ \\ & -m\cdot y_n^{\prime }& \le & x\\ \\ & x& \ge & (-m-1)\cdot (1-y_p^{\prime })+1\\ \\ & x& \le & (m+1)\cdot (1-y_n^{\prime })-1\\ \\ & y_p^{\prime }+y_n^{\prime }& =& 1\end{array}$$

连续（非精确）

常量

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

额外变量

$b_p\in [0,1]$ ：多项式为正数的归一化值。

$b_n\in [0,1]$ ：多项式为负数的归一化值。

$y_p^{\prime }\in \{0,1\}$ ：多项式为真的逻辑值。

$y_n^{\prime }\in \{0,1\}$ ：多项式为假的逻辑值。

导出符号

$$y=y_p^{\prime }-y_n^{\prime }$$

数学模型

$$\begin{array}{rlrl}s.t.& x& =& -m\cdot b_n+m\cdot b_p\\ \\ & y_p& \ge & b_p\\ \\ & ϵ\cdot y_p& \le & b\\ \\ & y_n& \ge & b_n\\ \\ & ϵ\cdot y_n& \le & b\\ \\ & b_n+y_p& \le & 1\\ \\ & b_p+y_n& \le & 1\end{array}$$

即：

$$y=\{\begin{array}{ll}1,& x\ge ϵ\\ \\ 0,& x=0\\ \\ -1,& x\le -ϵ\end{array}$$

连续（精确）

导出符号

$$y=Ulp(x)$$

$Ulp(x)$ 可参考 一元分段线性函数，使用的采样点为 $\{(min(x),-1)(-p,-1),(-p+ϵ,0),(0,0),(p-ϵ,0),(p,1),(max(x),1)\}$。

即：

$$y=\{\begin{array}{ll}1,& x\ge p\\ \\ 0,& -p<x<p\\ \\ -1,& x\le -p\end{array}$$

样例

离散

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 = IntVar("x")
x.range.leq(Int64.two)
x.range.geq(-Int64.two)
val bter = BalanceTernaryzationFunction(x, name = "bter")
val solver = ScipLinearSolver()

val model1 = LinearMetaModel()
model1.add(x)
model1.add(bter)
model1.minimize(bter)
val result1 = runBlocking { solver(model1) }
assert(result1.value!!.obj eq -Flt64.one)

val model2 = LinearMetaModel()
model2.add(x)
model2.add(bter)
model2.maximize(bter)
val result2 = runBlocking { solver(model2) }
assert(result2.value!!.obj eq Flt64.one)

val model3 = LinearMetaModel()
model3.add(x)
model3.add(bter)
model3.addConstraint(x geq Flt64.zero)
model3.minimize(bter)
val result3 = runBlocking { solver(model3) }
assert(result3.value!!.obj eq Flt64.zero)

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

val model5 = LinearMetaModel()
model5.add(x)
model5.add(bter)
model5.addConstraint(x leq 0)
model5.minimize(bter)
val result5 = runBlocking { solver(model5) }
assert(result5.value!!.obj eq -Flt64.one)

val model6 = LinearMetaModel()
model6.add(x)
model6.add(bter)
model6.addConstraint(x leq 0)
model6.maximize(bter)
val result6 = runBlocking { solver(model6) }
assert(result6.value!!.obj eq Flt64.zero)

连续

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.two)
val bter = BalanceTernaryzationFunction(x, name = "bter")
val solver = ScipLinearSolver()

val model1 = LinearMetaModel()
model1.add(x)
model1.add(bter)
model1.minimize(bter)
val result1 = runBlocking { solver(model1) }
assert(result1.value!!.obj eq -Flt64.one)

val model2 = LinearMetaModel()
model2.add(x)
model2.add(bter)
model2.maximize(bter)
val result2 = runBlocking { solver(model2) }
assert(result2.value!!.obj eq Flt64.one)

val model3 = LinearMetaModel()
model3.add(x)
model3.add(bter)
model3.addConstraint(x geq Flt64.zero)
model3.minimize(bter)
val result3 = runBlocking { solver(model3) }
assert(result3.value!!.obj eq Flt64.zero)

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

val model5 = LinearMetaModel()
model5.add(x)
model5.add(bter)
model5.addConstraint(x leq 0)
model5.minimize(bter)
val result5 = runBlocking { solver(model5) }
assert(result5.value!!.obj eq -Flt64.one)

val model6 = LinearMetaModel()
model6.add(x)
model6.add(bter)
model6.addConstraint(x leq 0)
model6.maximize(bter)
val result6 = runBlocking { solver(model6) }
assert(result6.value!!.obj eq Flt64.zero)

val model7 = LinearMetaModel()
model7.add(x)
model7.add(bter)
model7.addConstraint(x leq Flt64(0.3))
model7.maximize(bter)
val result7 = runBlocking { solver(model7) }
assert(result7.value!!.obj eq Flt64.one)

val model8 = LinearMetaModel()
model8.add(x)
model8.add(bter)
model8.addConstraint(x geq -Flt64(0.3))
model8.minimize(bter)
val result8 = runBlocking { solver(model8) }
assert(result8.value!!.obj eq -Flt64.one)

完整实现请参考：

• Kotlin

完整样例请参考：

• Kotlin