使用 R 找到具有四个未知数的两个非线性方程的决策边界

Question

Consider two equations, (1-b1)*(.4*Y1-5) and (1-b2)*(.4*Y2-5), where b1 and b2 are probabilities from 0 to 1 and b2 must always be larger than b1 and Y1 and Y2 can be any number from 50 to 100, but Y2 must always be larger than Y2. I am trying to find the decision boundary where these equations equal one another within the given constraints of b2 and Y2 using R.

I have tried uniroot and solve, however, it seems that uniroot can only be used when there is one unknown and solve requires a systems of linear equations.

Is there any function that can set two equations with four unknowns equal to one another to determine where the decision boundary is.

Answer 1

Since there are multiple variables to be solved, you can use optim instead of uniroot :

optim(c(0.5, 0.6, 75, 80),
    function(x) {
        b1 <- x[1L]
        b2 <- x[2L]
        Y1 <- x[3L]
        Y2 <- x[4L]

        if (b1 < 0 | b1 > 1 | b2 < 0 | b2 > 1 | b1 > b2)
            return(Inf)

        if (Y1 < 50 | Y1 > 100 | Y2 < 50 | Y2 > 100 | Y1 > Y2)
            return(Inf)

        abs((1-b1)*(.4*Y1-5) - (1-b2)*(.4*Y2-5))
    })

output:

$par
[1]  0.5098563  0.5917020 75.1616219 87.7225153

$value
[1] 3.52709e-08

$counts
function gradient 
     205       NA 

$convergence
[1] 0

$message
NULL