Ford vs. Ferrari

I recently was reminded to look again at Ignacio Sarmiento Barbieri's blog post
https://ignaciomsarmiento.github.io/2017/01/20/Quantile-and-OLS-Regression-with-Rmosek.html
that compares the speed of several implementations of quantile regression in Mosek with
that of the simplex implementation in my quantreg R package.  He finds that for problems
of modest size (n = 1000) the quantreg simplex implementation which is based on the 1972
algorithm of Barrodale and Roberts was substantial faster than any of the Mosek implementations.

However, in a more realistic econometric problem based on data from Josh Angrist (n = 97367, p = 5)
the simplex algorithm was quite a bit slower than the best of the Mosek implementations.  I wondered
how the interior point algorithms implemented in fortran in quantreg would perform.

Now, Mosek is clearly the Ferrari in this contest.  I've used Mosek in many different convex
optimization problems and always found it to be remarkably quick, responsive to special road
conditions and (especially) elegant in design.  So it requires a special sort of hubris to think
that the so-called Frisch-Newton implementations that were figuratively "made in my garage"
could compete, but lo and behold, here are the results:

Unit: milliseconds

                               expr       min        lq      mean    median

 qr.mosek.dual.rep(X, y, tau = tau)  346.9841  357.2982  365.9998  364.2814

        rq.fit.fnb(X, y, tau = tau)  193.7651  194.5455  202.1676  195.2470

       rq.fit.sfn(Xs, y, tau = tau)  330.3882  341.1280  365.0841  349.7058

         rq.fit.br(X, y, tau = tau) 7266.7237 7270.0682 7289.8437 7274.8538

        uq       max neval

  370.6428  462.1361   100

  206.7692  288.0764   100

  369.8432  451.3124   100

 7285.0870 7522.6136   100

The results for the basic (dense) Frisch-Newton appear as rq.fit.fnb while the sparse version of this
algorithm appear as rq.fit.sfn and the simplex version is rq.fit.br.