Attic method removal

In the strategy proposed for the Attic method, we mentioned that an inequality constraint cannot be removed from the matrix J even though Xj < 0 until a vertex is... 

In the Attic method, after preprocessing during which superfluous variables and constraints are removed and the bound constraints are calculated for each variable, the problem is formulated as follows ... 

The Attic method is based on the idea of introducing one inequality constraint at a time, selecting each one from the most promising ones. Once a vertex of nv constraints is set up, more inequality constraints are simultaneously removed when opportune. From a certain point of view, this strategy is similar to the stepwise method of building the best model in a linear regression problem (Vol. 2 -Buzzi-Ferraris and Manenti, 2010b). A forward method is used to insert constraints and a backward method to remove them. 

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 third problem is knowing which constraints must be kept active at a certain iteration. In line with the philosophy of the Attic method, only the equality constraints are always considered active. The inequality and bound constraints are gradually added one by one until saturation is achieved (no other constraints can be added). If the constraints inserted fulfill the KKT conditions, the solution is achieved otherwise, the inequality constraints or bounds with 1 < 0 are removed. 

When the working point is a degenerate attic vertex, the procedure continues for a limited number of iterations even though the movement between two adjacent iterations is null. Note that, at each iteration, there are many more opportunities to exit the degenerate point than with the Simplex method more constraints with Aj < 0 can be simultaneously removed, and more constraints with A > 0 can also be removed. The procedure could be suddenly stopped to prevent cycling even though the assigned number of iterations is not yet accompUshed. At this point, the overall procedure moves toward the last phase. 

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