# Why scaling and geometry is relevant

### Why scaling and geometry is relevant

This section provides a simple example of how scaling problems can slow down
problem solving and, in extreme cases, result in unexpected
answers. Consider the problem:

and let be a diagonal matrix where . In theory, solving should be equivalent to solving the related problem :

However, in practice, the two models behave very differently. To demonstrate this, we use a simple script

`rescale.py`that randomly rescales the columns of the model. Let's consider the impact of rescaling on the problem

`pilotnov.mps.bz2`. Solving the original problem gives the following output:

Optimize a model with 975 rows, 2172 columns and 13057 nonzeros Coefficient statistics: Matrix range [3e-06, 9e+06] Objective range [3e-03, 1e+00] Bounds range [6e-06, 7e+04] RHS range [1e-05, 4e+04] Warning: Model contains large matrix coefficient range Consider reformulating model or setting NumericFocus parameter to avoid numerical issues. Presolve removed 254 rows and 513 columns Presolve time: 0.01s Presolved: 721 rows, 1659 columns, 11454 nonzeros Iteration Objective Primal Inf. Dual Inf. Time 0 -3.2008682e+05 1.435603e+05 0.000000e+00 0s 1137 -4.4972762e+03 0.000000e+00 0.000000e+00 0s Solved in 1137 iterations and 0.13 seconds Optimal objective -4.497276188e+03 Kappa: 1.949838e+06

Note the log message regarding the matrix coefficient range in the log (which in this case shows a range of [3e-06, 9e+06]).

If we run `rescale.py -f pilotnov.mps.bz2 -s 1e3` (randomly rescaling
columns up or down by as much as ), we obtain:

Optimize a model with 975 rows, 2172 columns and 13057 nonzeros Coefficient statistics: Matrix range [5e-09, 1e+10] Objective range [2e-06, 1e+03] Bounds range [5e-09, 6e+07] RHS range [1e-05, 4e+04] Warning: Model contains large matrix coefficient range Consider reformulating model or setting NumericFocus parameter to avoid numerical issues. Presolve removed 100 rows and 255 columns Presolve time: 0.00s Presolved: 875 rows, 1917 columns, 11899 nonzeros Iteration Objective Primal Inf. Dual Inf. Time 0 -6.2117921e+32 7.026405e+31 6.211792e+02 0s Extra 2 simplex iterations after uncrush 1166 -4.4972762e+03 0.000000e+00 0.000000e+00 0s Solved in 1166 iterations and 0.15 seconds Optimal objective -4.497276188e+03 Kappa: 2.341493e+18

This time, the optimization process takes a more
iterations, and also, we get an extra warning:

`Extra 2 simplex iterations after uncrush`,

This indicates that extra simplex iterations were performed on
the unpresolved model. Also, note the very large value for ` Kappa`; its meaning will be discussed in this section.

If we run `rescale.py -f pilotnov.mps.bz2 -s 1e6`, we
obtain:

Optimize a model with 975 rows, 2172 columns and 13057 nonzeros Coefficient statistics: Matrix range [5e-12, 1e+13] Objective range [2e-09, 1e+06] Bounds range [5e-12, 5e+10] RHS range [1e-05, 4e+04] Warning: Model contains large matrix coefficient range Warning: Model contains large bounds Consider reformulating model or setting NumericFocus parameter to avoid numerical issues. Presolve removed 103 rows and 252 columns Presolve time: 0.01s Presolved: 872 rows, 1920 columns, 11900 nonzeros Iteration Objective Primal Inf. Dual Inf. Time 0 -6.4093202e+34 7.254491e+31 6.409320e+04 0s Extra 151 simplex iterations after uncrush 1903 -4.4972762e+03 0.000000e+00 0.000000e+00 0s Solved in 1903 iterations and 0.23 seconds Optimal objective -4.497276188e+03 Warning: unscaled primal violation = 0.171778 and residual = 0.00142752 Kappa: 5.729068e+12

Now we get a much larger number of extra simplex iterations,
and more troublingly, we get a warning about the quality of the
resulting solution:

`Warning: unscaled primal violation = 0.171778
and residual = 0.00142752`,

This message indicates that the solver had trouble finding a solution
that satisfies the default tolerances.

Finally, if we run `rescale.py -f pilotnov.mps.bz2 -s 1e8`,
we obtain:

Optimize a model with 975 rows, 2172 columns and 13054 nonzeros Coefficient statistics: Matrix range [3e-13, 7e+14] Objective range [2e-11, 1e+08] Bounds range [5e-14, 1e+13] RHS range [1e-05, 4e+04] Warning: Model contains large matrix coefficient range Warning: Model contains large bounds Consider reformulating model or setting NumericFocus parameter to avoid numerical issues. Presolve removed 79 rows and 242 columns Presolve time: 0.00s Solved in 0 iterations and 0.00 seconds Infeasible modelIn this case, the optimization run terminates almost instantly, but with the unexpected

`Infeasible`result.

As you can see, as we performed larger and larger rescalings, we
continued to obtain the same optimal value, but there were clear
signs that the solver struggled. We see warning messages, as well
increasing iteration counts, runtimes, and `Kappa` values.
However, once we pass a certain rescaling value, the solver is no longer able to
solve the model and instead reports that it is `Infeasible`.

Note that this is not a bug in Gurobi. It has to do with changing the meaning of numbers depending on their range, the use of fixed tolerances, and in the changing geometry of the problem due to scaling. We will discuss this topic further in a later section.