This is not, per se, very useful, but sorting a parameter inside a MIP model is not very easy for MIP solvers. Obviously, if you need a sorted parameter in the model, it is better to use a sorting algorithm. But useless models can still be interesting.
I use:
Input: a 1-dimensional parameter with values.
Output: a 1-dimensional variable with the values sorted in ascending order.
We can implement this with a permutation matrix \(\color{darkred}X\), which is a permuted identity matrix. In a MIP context, this becomes a binary variable with some assignment constraints.
| MIP Model for sorting \(p_i\) |
|---|
| \[\begin{align}\min\>&\color{darkred}z=0\\& \sum_i \color{darkred}x_{i,j} = 1&&\forall j\\ & \sum_j \color{darkred}x_{i,j} = 1&&\forall i\\ & \color{darkred}y_i = \sum_j \color{darkred}x_{i,j}\cdot\color{darkblue}p_j\\& \color{darkred}y_i \ge \color{darkred}y_{i-1}\\ & \color{darkred}x_{i,j} \in \{0,1\}\end{align}\] |
If we want to sort \(n\) values, this model has \(n^2\) binary variables.
One possible improvement is to add the extra constraint \[\sum_i \color{darkred}y_i = \sum_i \color{darkblue}p_i\]
Let's compare a commercial solver (Cplex) with an open-source one (HiGHS). I use here \(n=50,100,200\). Model 1 is without the extra constraint while model 2 includes the extra constraint. The data \(\color{darkblue}p_i\) is random. A time limit of 1,000 seconds was used. Otherwise, the solvers used default settings.
---- 96 PARAMETER results solver performance status time nodes iterations cplex.n=50 .model1 Optimal 2.970000E-12408 cplex.n=50 .model2 Optimal 2.500000E-12151 cplex.n=100.model1 Optimal 414055 cplex.n=100.model2 Optimal 27092 cplex.n=200.model1 Optimal 22730431684124 cplex.n=200.model2 Optimal 1810155 highs.n=50 .model1 Optimal 2 highs.n=50 .model2 Optimal 2 highs.n=100.model1 TimeLimit 100195874955336 highs.n=100.model2 TimeLimit 1000152215373228
- Cplex is much better on the larger models
- HiGHS is very promising on the \(n=50\) model: it solves this in the presolver (zero nodes, zero LP iterations). But it runs out of steam for large models. I would expect that if the presolver can solve the problem for \(n=50\), it would also be able to do this for \(n=100\). As usual, my predictions are not that good.
- The extra constraint is useful for Cplex, especially for the larger models. Both in terms of time and LP iterations.
The Xpress solver was complaining about numerical issues:
Numerical issues encountered:Dual failures : 831 out of 5345 (ratio: 0.1555)Singular bases : 1 out of 50821 (ratio: 0.0000)
I am a little bit surprised by this. Scaling is the most notorious reason for numerical problems in LP/MIP models. Since this model does not have big-M constraints, scaling is excellent. Of course, I don't how the cuts added by the solver look like. I don't really know what to do about this. Maybe the absence of an objective is a problem.
Sorting a variable
Here, we sorted a parameter. That is relatively easy. Sorting a variable inside a model is more of a challenge. The constraint \[\color{darkred}y_i = \sum_j \color{darkred}x_{i,j}\cdot\color{darkred}p_j\] is now quadratic. The products \(\color{darkred}x_{i,j}\cdot\color{darkred}p_j\) (a binary times a continuous variable) can be linearized, but that will give us some big-M constraints. Indicator constraints are an option if no good bounds on \(\color{darkred}p_j\) are available. This is again a case where using quadratic formulations can make the modeling way easier.
Here are some formulations:
- Non-convex Quadratic: \[ \color{darkred}y_i = \sum_j \color{darkred}x_{i,j}\cdot\color{darkred}p_j\] You need a global quadratic solver for this.
- Linearization using binary variables: the general case. Assume \(\color{darkred}p_j\in [\color{darkblue}\ell_j,\color{darkblue}u_j]\).We can linearize the products \(\color{darkred}w_{i,j}=\color{darkred}x_{i,j}\cdot\color{darkred}p_j\) as follows: \[\begin{align} & \color{darkred}y_i = \sum_j \color{darkred}w_{i,j} \\ & \color{darkblue}\ell_j \cdot \color{darkred}x_{i,j} \le \color{darkred}w_{i,j} \le \color{darkblue}u_j \cdot \color{darkred}x_{i,j}\\ & \color{darkred}p_j - \color{darkblue}u_j\cdot (1-\color{darkred}x_{i,j}) \le \color{darkred}w_{i,j} \le \color{darkred}p_j - \color{darkblue}\ell_j\cdot (1-\color{darkred}x_{i,j}) \\ & \color{darkred}w_{i,j} \in [\min\{0,\color{darkblue}\ell_j\},\max \{0,\color{darkblue}u_j\}]\end{align}\]
- If \(\color{darkred}p_j\in [0,\color{darkblue}u_j]\), things become a bit simpler: \[\begin{align} & \color{darkred}y_i = \sum_j \color{darkred}w_{i,j} \\ & \color{darkred}w_{i,j} \le \color{darkblue}u_j \cdot \color{darkred}x_{i,j}\\ & \color{darkred}p_j - \color{darkblue}u_j\cdot (1-\color{darkred}x_{i,j}) \le \color{darkred}w_{i,j} \le \color{darkred}p_j \\ & \color{darkred}w_{i,j} \in [0,\color{darkblue}u_j]\end{align}\]
- Using indicator constraints, we can write: \[\begin{align}& \color{darkred}y_i = \sum_j \color{darkred}w_{i,j} \\ & \color{darkred}x_{i,j}=0 \implies \color{darkred}w_{i,j}=0 \\ & \color{darkred}x_{i,j}=1 \implies \color{darkred}w_{i,j}=\color{darkred}p_j \end{align}\]
- We can also use a SOS1 formulation. \[\begin{align}& \color{darkred}y_i = \sum_j \color{darkred}w_{i,j} \\ & \color{darkred}x_{i,j}+\color{darkred}s^1_{i,j}=1\\ & \color{darkred}w_{i,j}+\color{darkred}s^2_{i,j}=0 \\ & \color{darkred}w_{i,j}+\color{darkred}s^3_{i,j}=\color{darkred}p_j \\ & (\color{darkred}s^1_{i,j}, \color{darkred}s^2_{i,j}) \in SOS1 \\ & (\color{darkred}x_{i,j},\color{darkred}s^3_{i,j})\in SOS1 \end{align}\] The slack variables are free. This formulation has a lot of SOS1 sets, each one having two members.
As in the previous section, we can add the extra constraint \[\sum_i \color{darkred}y_i = \sum_i \color{darkred}p_i\]
Here are some timings of a model using \(n=50\):
---- 203 PARAMETER resultssolver performance status obj time nodes iterations Baron.quad Optimal -5108761 Baron.quad+extra Optimal -51013091 Cplex.bin Optimal -51013075311421599612 Cplex.bin+extra Optimal -5106914931815245649 Cplex.indic Optimal -5101098117019969241140 Cplex.indic+extra Optimal -5101442097647843900 Cplex.sos1 TimeLimit -414361195960115280454 Cplex.sos1+extra Optimal -5103362214940351270681
The solution times fluctuate quite a bit. The extra constraint usually performs much better (the global MINLP solver Baron is the exception here).
The quadratic formulation is my favorite. It is succinct and compact, provides all the information the solver needs, and is easy to read. My argument here is that quadratic terms are useful as a modeling device. The modeling tool or solver should handle the linearization. These linearizations can be non-trivial.