## Thursday, December 14, 2017

### Permuted Matrix Balancing

In [1] the following problem is posed:

#### Problem

I am trying to generate a matrix of numbers with 7 rows and 4 columns. Each row must sum to 100 and each column must have an even spread (if permitted) between a min and max range (specified below).
Goal:
       C1      C2    C3    C4   sum   range
1     low                      100    ^
2     ..                              |
3     ..                              |
4     ..                              |
5     ..                              |
6     ..                              |
7     high                            _

c1_high = 98
c1_low = 75
c2_high = 15
c2_low = 6
c3_high = 8
c3_low = 2
c4_low = 0.05
c4_high =0.5
In addition to this, i need the spread of each row to be as linear as possible, though a line fitted to the data with a second order polynomial would suffice (with an r^2 value of >0.98).
Note: this description seems to hint the values in the columns must be ordered from $$L_j$$ to $$U_j$$. That is actually not the case: we don't assume any ordering.

#### Solution Outline

This problem somewhat resembles a matrix balancing problem. Let's see how we can model this.

One approach to attack this problem is:
##### Step 1: Create data with perfect spread
It is easy to generate data with perfect spread wile ignoring that the row sums are 100. We just start at $$L_j$$ and increase a by fixed step size:

----     40 PARAMETER L  low

c1 75.000,    c2  6.000,    c3  2.000,    c4  0.050

----     40 PARAMETER U  high

c1 98.000,    c2 15.000,    c3  8.000,    c4  0.500

----     40 PARAMETER init  initial matrix

c1          c2          c3          c4      rowsum

r1       75.000       6.000       2.000       0.050      83.050
r2       78.833       7.500       3.000       0.125      89.458
r3       82.667       9.000       4.000       0.200      95.867
r4       86.500      10.500       5.000       0.275     102.275
r5       90.333      12.000       6.000       0.350     108.683
r6       94.167      13.500       7.000       0.425     115.092
r7       98.000      15.000       8.000       0.500     121.500


This step is just data manipulation.
##### Step 2: Permute values in each column
The second step is to permute the values within a column to achieve a solution that has a close match to the row sums. This gives:

----     75 VARIABLE y.L  after permutation

c1          c2          c3          c4      rowsum

r1      86.500      10.500       5.000       0.275     102.275
r2      94.167       7.500       3.000       0.125     104.792
r3      98.000       6.000       2.000       0.050     106.050
r4      90.333       9.000       4.000       0.200     103.533
r5      82.667      12.000       6.000       0.350     101.017
r6      75.000      15.000       8.000       0.500      98.500
r7      78.833      13.500       7.000       0.425      99.758


We are closer to the row sums as expected.

One way to model a permutation of a column vector $$a$$ inside an optimization model is to find a square permutation matrix $$P$$ [2] with

\begin{align}&y_i = \sum_k P_{i,k}a_k\\ &\sum_k P_{i,k}=1&\forall i \\&\sum_i P_{i,k} = 1&\forall k\\& P_{i,k}\in\{0,1\}\end{align}

A permutation matrix has exactly one one in each row and in each column. The rest of the elements are zero. One could say a permutation matrix is a (row or column) permuted identity matrix. The constraints can also be interpreted as assignment constraints (related to the assignment problem).

##### Step 3: Add minimal relative deviations
The final step is to find small deviations from $$y$$ such that the row sums are exactly 100.

----     75 VARIABLE x.L  final values

c1          c2          c3          c4      rowsum

r1      84.265      10.467       4.993       0.275     100.000
r2      89.460       7.431       2.984       0.125     100.000
r3      92.057       5.912       1.980       0.050     100.000
r4      86.863       8.949       3.988       0.200     100.000
r5      81.668      11.985       5.997       0.350     100.000
r6      76.473      15.022       8.005       0.500     100.000
r7      79.071      13.503       7.001       0.425     100.000

----     75 VARIABLE d.L  deviations

c1          c2          c3          c4

r1      -2.235      -0.033      -0.007 -2.25855E-5
r2      -4.707      -0.069      -0.016 -4.75702E-5
r3      -5.943      -0.088      -0.020 -6.00626E-5
r4      -3.471      -0.051      -0.012 -3.50779E-5
r5      -0.999      -0.015      -0.003 -1.00932E-5
r6       1.473       0.022       0.005 1.489155E-5
r7       0.237       0.003 7.931220E-4 2.399194E-6


For this I used a relative measure: minimize the sum of squared relative deviations:

$\min \sum_{i,j} \left( \frac{d_{i,j}}{\mu_j} \right)^2$

where $$\mu_j = (L_j +U_j)/2$$. This is to ensure that the deviations are distributed according to the average magnitude of a column. This is to make sure column 1 gets larger deviations than column 4.

#### A complete model

I believe we need to do steps 2 and 3 simultaneously to find a global optimal solution. Here is an attempt:

This is a convex MIQP model, so we can solve this with high-performance solvers like Cplex and Gurobi.

#### Notes

1. We could split the model in two phases (step 2 and step 3 in the above description). I am not sure how this would impact the optimal solution. Also it would not really simplify things.
2. This model can be linearized by using a different objective: minimize the sum of the absolute values of the (relative) deviations:
$\min\>\sum_{i,j}\left| \frac{d_{i,j}}{\mu_j} \right|$
You would end up with a linear MIP model. Possible formulations are variable splitting
\begin{align}\min \> & \sum_{i,j} v_{i,j}^{\oplus} + v_{i,j}^{\ominus}\\&v_{i,j}^{\oplus} - v_{i,j}^{\ominus} = \frac{d_{i,j}}{\mu_j}\\&v_{i,j}^{\oplus}, v_{i,j}^{\ominus}\ge 0 \end{align}
or bounding:
\begin{align}\min\>&\sum_{i,j} v_{i,j}\\&-v_{i,j} \le \frac{d_{i,j}}{\mu_j}\le v_{i,j}\end{align}
3. Instead of dividing by the average column value $$\mu_j$$ we can divide by the "real" value $$y_{i,j}$$. Unfortunately, that would make this a general MINLP model. Note that we should "protect" the division, e.g. by the bound $$y_{i,j}\ge L_j$$.

#### References

1. Algorithm for generation of number matrix with specified boundaries, https://stackoverflow.com/questions/47754210/algorithm-for-generation-of-number-matrix-with-specified-boundaries
2. Permutation Matrix, https://en.wikipedia.org/wiki/Permutation_matrix