Equality Lagrange parameters

The seminal idea on which active set methods are based is rather simple and is the same, albeit with several variants, as the one adopted in the Attic method as well as in the Simplex method for linear programming starting from a point where certain constraints are active (all the equality constraints and some inequality constraints), we search for the solution to this problem as if all the constraints are equalities. During the search, it, however, may be necessary to insert other inequality constraints that were previously passive and/or remove certain active inequality constraints as they are considered useless based on their Lagrange parameters. The procedure continues until KKT conditions are fulfilled. 

The above constrained parameter estimation problem becomes much more challenging if the location where the constraint must be satisfied, (xo,yo), is not known a priori. This situation arises naturally in the estimation of binary interaction parameters in cubic equations of state (see Chapter 14). Furthermore, the above development can be readily extended to several constraints by introducing an equal number of Lagrange multipliers. 

The equilibrium order parameters X g and rjeq minimize AF subject to any system constraints. Supposing that the system s composition is fixed, the method of Lagrange multipliers leads to a common-tangent construction for AF with respect to XB—or equivalently, equality of chemical potentials of both A and B. Two compositions, Xjj and X +, will coexist at equilibrium for average compositions XB in the composition range Xe < XB < -X Bq+ if they satisfy... 

Recall that in the moment approach, each phase a is parameterized by Lagrange multipliers kf for the original moments (the ones appearing in the excess free energy of the system) and the fraction of system volume that it occupies. If extra moments are used, there is one additional Lagrange multiplier A, for each of them these are common to all phases. These parameters have to be chosen such that the pressure (44) and the moment chemical potentials nt given by Eq. (42) are equal in all phases. Furthermore, the (fractional) phase volumes v have to sum to one, and the lever rule has to be satisfied for all moments (both original and extra) ... 

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