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I want two(2) numbers $\phi_{1}, \quad \phi_{2}$ such that the following five(5) conditions will be satisfied using R:

  1. $ \phi_{1} + \phi_{2} < 1 $
  2. $ \phi_{1} - \phi_{2} < 1 $
  3. $ -1 < \phi_{1} < 1 $
  4. $ -1 < \phi_{2} < 1 $
  5. $\phi_{1}$ and $\phi_{2}$ are in one(1) decimal place
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  • 3
    $\begingroup$ $$\phi_1=\phi_2=0$$ $\endgroup$ – Xi'an Jan 12 at 12:01
  • $\begingroup$ This is called a linear program and the subject that addresses its solution is called linear programming. The particular problem of finding a solution is called finding a feasible solution. $\endgroup$ – whuber♦ Jan 12 at 14:09
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Brute force, not much thinking: $\phi_1$ and $\phi_2$ are both between -1 and 1 and they are given to one decimal place. That makes them discrete and we can simply test all possible combinations of $\phi_1$ and $\phi_2$.

phi1 <- seq(-.9, .9, .1)
phi2 <- seq(-.9, .9, .1)
grid <- expand.grid(phi1 = phi1, phi2 = phi2)

grid$condition1 <- grid$phi1 + grid$phi2 < 1
grid$condition2 <- grid$phi1 - grid$phi2 < 1

grid$valid <- grid$condition1 & grid$condition2
valid.combinations <- grid[which(grid$valid),]

plot(grid$phi1, grid$phi2, xlim = c(-1, 1), ylim = c(-1,1), col = "grey", ,
      xlab = expression(phi[1]), ylab = expression(phi[2]))
points(valid.combinations$phi1, valid.combinations$phi2, pch = 16)

enter image description here

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