Scaling

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Changing the units of measurement, usually for the numerical stability of an algorithm. The variables are transformed as LaTeX: \textstyle x' = \mbox{S}x, where LaTeX: \textstyle \mbox{S} = \mbox{diag}(s_j). The diagonal elements are the scale values, which are positive: LaTeX: \textstyle s_1, \dots , s_n > 0. Constraint function values can also be scaled. For example, in an LP, the constraints LaTeX: \textstyle \mbox{A}x = b, can be scaled by LaTeX: \textstyle \mbox{RA} x = \mbox{R} b , where LaTeX: \textstyle \mbox{R} = \mbox{diag}(r_i) such that LaTeX: \textstyle r > 0. (This affects the dual values.) Some LP scaling methods simply scale each column of A by dividing by its greatest magnitude (null columns are identified and removed).


Example A column scaling A row scaling
10x + 100y = 500
–30x + .3y = .2
.3333x + y = 500
–x + .003y = .2
x + 10y = 50
–300x + 3y = 2


Another method is logarithmic scaling, which scales by the logarithm of the greatest magnitude. More sophisitcated methods are algorithmic, taking both row and column extremes into account.


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