I found it to be an interesting exercises to write an optimization model using the Optimization Programming Language (OPL) provided by IBM ILOG CPLEX.

The quantile regression OPL code is provided in the sub-directory OPL in MathematicaForPrediction project at GitHub. The code can use only first order B-splines. The number of knots and the quantile are parameters to be specified. The reason to have only first order splines is because OPL provides only one type of piecewise functions — linear piecewise functions. OPL piecewise functions are somewhat awkward to interpret, but relatively easy to specify. (I will update this post with Mathematica code to parse and interpret them.) Since, of course, OPL’s `piecewise{...}`

can be used to specify step functions I could have programmed zero order splines, but that would be only useful for educational purposes.

The code is short:

```
```tuple Point {

float x;

float y;

};

// Data points provided by an external file

{Point} data = ...;

// Number of knots parameter

int k = 7;

// Quantile parameter

float q = 0.75;

float dataMin = min(d in data) d.x;

float dataMax = max(d in data) d.x;

float cLen = ( dataMax - dataMin ) / (k-1);

dvar float+ u[data];

dvar float+ v[data];

dvar float+ bs[1..k];

minimize sum(d in data) q*u[d] + sum(d in data) (1-q)*v[d];

subject to {

` forall ( d in data ) {`

(sum ( i in 1..k ) bs[i] * (piecewise{ 0-> dataMin + (i-2)*cLen; 1/cLen -> dataMin + (i-1)*cLen; -1/cLen -> dataMin + i*cLen; 0}( dataMin + (i-2)*cLen, 0) d.x) ) + u[d] - v[d] == d.y;

}

}

The code mentioned earlier provides print-outs of the piecewise, B-spline basis functions and the weights found for them. (The weighted sum of the B-spline basis functions gives the regression quantile.)