[Clp] Interpretation of solution variables beyond bounds

[Lou Hafer]

Alex,

> There remain doubts about effect of scaling on tolerance. Can scaling move
> actual (unscaled) variable beyond of the limit of configured tolerance?

    If memory serves, clp confirms the scaled solution using the unscaled
    problem.  If the result is not within tolerance, the basis from the
    scaled solution is used as the starting point for further pivots.

> Does scaling affect dual infeasibilities?

    Scaling can alter all matrix coefficients, as well as row and column
    bounds.  You might want to review the math involved.  If nothing else is
    handy for you, there's an explanation in the dylp documentation.

    Keep in mind that computer floating point arithmetic is finite precision
    and inherently inaccurate to some degree.  If absolute accuracy to n
    fractional digits is critically important for your application, you
    might want to look at the exact arithmetic capability in the glpk
    solver.  It's not something you want to use to solve a problem from
    scratch, but it might prove interesting as a check.  The result, of
    course, is only as good as the inputs.  Most often, input coefficient
    accuracy is the limiting factor for problems involving real-world data.
    Default primal and dual tolerances are set with this in mind.  Problems
    derived from mathematical constructions are a different story.

    						Lou