How to formulate case distinctions in AMPLs objective function?

Question

This is my first real optimisation problem I formulated and now trying to solve by using AMPL.

The following objective function is from a linear 0-1 LP means all variables $x_i^b\in\{0,1\}$, with $i\in[1,n]$ and $b$ referring to the type of the node, which means $0$ and $1$ say a node is or is not of a certain type.

The objective function is as follows with $A$ a set of nodes, $N[a]$ a set of all neighbours of $a$ including $a$, and $n$ a given number of types:

$$\min - \sum_{a\in A}f\left(\frac1n\sum\limits_{i=1}^nf\left(\sum\limits_{b\in N[a]}x_i^b\right)\right)$$

with

$$ f:\mathbb{R}\mapsto\{0,1\}\\ f(x):=\begin{cases}0,&x<1\\1,&x\geq 1\end{cases} $$

I already heard that AMPL doesn't support the definition of functions/methods and that I have to use the Big M method to create this objective function. I couldn't really figure out yet how to use the method to replace my case distinctions...

Answer 1

Let $\rho$ be some small value. \begin{align}M \times f(x) &\geq x - 1 + \rho\\M \times (1 - f(x)) &\geq 1 - x - \rho\end{align}

Here is the small working code in Python pulp

import pulp as pl

prob = pl.LpProblem("Problem", pl.LpMinimize)
x = 1
f = pl.LpVariable("f_{0}", 0, 1, pl.LpBinary)
prob += f
M = 100
prob += M * f >= x - 1 + 0.001
prob += M * (1 - f) >= 1 - x - 0.0001

print(prob)
prob.solve()

for v in prob.variables():
    print(v.name, "=", v.varValue)