1Blurb::
2How to scale each linear equality constraint
3
4Description::
5Each string in \c linear_equality_scale_types indicates the scaling
6type for each linear equality constraint. They only have effect when
7the associated method specifies \c scaling.
8
9The options are:
10
11\li <tt>'value'</tt> - characteristic value. If this is chosen, then
12    \ref variables-linear_equality_scales must be specified; 'value'
13    is assumed if scales are given without a \c scale_types
14
15\li <tt>'auto'</tt> - automatic scaling.
16
17If a single string is specified it will apply to all linear equality
18constraints. Otherwise the number of strings specified should be equal
19to the number of linear equalities.
20
21Scaling for linear constraints is
22applied \e after any continuous variable scaling.
23
24For example, for
25variable scaling on continuous design variables x: \f[ \tilde{x}^j =
26\frac{x^j - x^j_O}{x^j_M} \f] we have the following system for linear
27equality constraints \f[ a_L \leq A_i x \leq a_U \f] \f[ a_L \leq
28A_i \left( \mathrm{diag}(x_M) \tilde{x} + x_O \right) \leq a_U \f] \f[
29a_L - A_i x_O \leq A_i \mathrm{diag}(x_M) \tilde{x} \leq a_U - A_i x_O
30\f] \f[ \tilde{a}_L \leq \tilde{A}_i \tilde{x} \leq \tilde{a}_U \f]
31and user-specified or automatically computed scaling multipliers are
32appplied to this final transformed system, which accounts for
33continuous design variable scaling. When automatic scaling is in use
34for linear constraints they are linearly scaled by a computed
35characteristic value, but not affinely to [0,1].
36
37See the scaling information under specific methods, e.g.,
38\c method-*-scaling for details on how to use this keyword.
39
40Topics::	linear_constraints
41Examples::
42Theory::
43Faq::
44See_Also::
45