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