Modeling surprises

#-------------------------------------------# funny behavior when using a != constraint# in a PuLP model#-------------------------------------------import pulp 

model = pulp.LpProblem("notequal")
x = pulp.LpVariable("x",lowBound=0,cat=pulp.LpInteger)
y = pulp.LpVariable("y",lowBound=0,cat=pulp.LpInteger)

model += x+y     # obj
model += pulp.LpConstraint(x != y) 

print(model)
# this is an alternative debugging tool: #    model.writeLP("/tmp/notequal.lp")# note that#    model += x != y # produces a different model!!

notequal:
MINIMIZE
1*x + 1*y + 0
SUBJECT TO
_C1:0 = -1

VARIABLES
0<= x Integer
0<= y Integer

The underlying issue is that operator overloading (redefining what an operator does for certain arguments) is very difficult to make bulletproof. Another example of this problem is shown in [2]. 

If a modeling tool wants to implement \(\color{darkred}x\ne \color{darkred}y\) for integer variables  \(\color{darkred}x,\color{darkred}y\) , it would need to generate something like : \(\color{darkred}x \le \color{darkred}y-1 \mathbf{\ or\ } \color{darkred}x \ge \color{darkred}y+1\). In a MIP model that could look like \[\begin{align}&\color{darkred}x\le\color{darkred}y-1+\color{darkblue}M_1\cdot\color{darkred}\delta\\&\color{darkred}x\ge\color{darkred}y+1-\color{darkblue}M_2\cdot(1-\color{darkred}\delta)\\&\color{darkred}\delta\in\{0,1\} \end{align}\] where \(\color{darkblue}M_1\), \(\color{darkblue}M_2\) are judiciously chosen large constants. If \(\color{darkred}x \in \{0,1,\dots,\color{darkblue}U_x\}\), \(\color{darkred}y \in \{0,1,\dots,\color{darkblue}U_y\}\), then the tightest values would be \(\color{darkblue}M_1 := 1+\color{darkblue}U_x\), \(\color{darkblue}M_2 := 1+\color{darkblue}U_y\). If no good upperbounds \(\color{darkblue}U\) are available, we can use either indicator constraints \[\begin{align}&\color{darkred}\delta=0\implies\color{darkred}x\le\color{darkred}y-1\\&\color{darkred}\delta=1\implies \color{darkred}x\ge \color{darkred}y+1\\&\color{darkred}\delta \in \{0,1\}\end{align}\] or SOS1 variables: \[\begin{align}&\color{darkred}x\le\color{darkred}y-1+\color{darkred}s_1\\& \color{darkred}x\ge \color{darkred}y+1 - \color{darkred}s_2 \\ & \color{darkred}s_1,\color{darkred}s_2\ge 0 \\& \{\color{darkred}s_1,\color{darkred}s_2\} \in \mathbf{SOS1} \end{align}\] depending on the capabilities of the modeling tools and solvers that are used.

The modeling language Minizinc can do the binary-variable transformation automatically when bounds are available. My example model looks like:

var 0..5: x;
var 0..7: y;

constraint x != y;

solve minimize x+y;

\ENCODING=ISO-8859-1
\Problem name: MIPCplexWrapper

Minimize
 obj1: X_INTRODUCED_0_
Subject To
 p_lin_0: x - y + 6 X_INTRODUCED_7_ <= 5
 p_lin_1: - x + y - 8 X_INTRODUCED_7_ <= -1
 p_lin_2: x + y - X_INTRODUCED_0_  = 0
Bounds
 0<= x <= 50<= y <= 70<= X_INTRODUCED_0_ <= 120<= X_INTRODUCED_7_ <= 1
Generals
 x  y  X_INTRODUCED_0_  X_INTRODUCED_7_
End

\[\begin{align}\min\>&\color{darkred}z \\ & \color{darkred}x \le \color{darkred}y - 1 + \color{darkblue}M_1 \cdot (1-\color{darkred}\delta) \\ &\color{darkred}x \ge \color{darkred}y + 1 - \color{darkblue}M_2 \cdot \color{darkred}\delta \\ & \color{darkred}z = \color{darkred}x+\color{darkred}y\end{align}\]