Another very small but very difficult global NLP model

The goal of this exercise is to fill a square area \([0,250]\times[0,100]\) with 25 circles. The model can choose the \(x\) and \(y\) coordinates of the center of each circle and the radius. So we have as variables \(\color{darkred}x_i\), \(\color{darkred}y_i\), and \(\color{darkred}r_i\). The circles placed inside the area should not overlap. The objective is to maximize the total area covered. 

A solution is:

Modeling surprises

Here is an example where the PuLP modeling tool goes berserk.

In standard linear programming, only \(\ge\), \(=\) and \(\le\) constraints are supported. Some tools also allow \(\ne\), which for MIP models needs to be reformulated into a disjunctive constraint. Here is an attempt to do this in PuLP [1]. PuLP does not support this relational operator in its constraints, so we would expect a meaningful error message.

Rounding inside an optimization model

In [1], the question was asked: how can I round to two decimal places inside an optimization model? I.e., \[\color{darkred}y_{i,j} = \mathbf{round}(\color{darkred}x_{i,j},2)\] To get this off my chest first: I have never encountered a situation like this. Rounding to two decimal places is more for reporting than something we want inside model equations. Given that, let me look into this modeling problem a bit more as an exercise.