Linear Attic method

The BzzFactorizedGaussAttic name comes from the fact that the first time it was used to solve a linear programming problem with the Attic method (see Chapter 10). The BzzFactorizedGaussAttic class includes the default constructor only. An object from this class is initialized using as the argument the matrix of the system and its right-hand side terms. 

The concept on which the Attic method is based is that any feasible point (vertex or nonvertex) sees the solution (it can be joined to the solution by a line) as the space is convex for all the linear programming problems (Buzzi-Ferraris, 2011b). 

The Attic method is based on the necessary conditions by Karush, Kuhn, and Tucker (KKT) applied to a problem, which is linear in its objective function and constraints. 

The Simplex and Attic methods can be seen as different strategies of active set methods applied to the special problem of linear function and linear constraints. 

The Attic method tries to exploit the fact that the ratio of the number of vertices to the number of constraints for linear programming problems is usually very large. 

For example, the ratio is 2 /nv for this class of problems since the number of constraints 2ny linearly increases with ttv- It is, therefore, suitable to consider a vertex as the intersection of ny constraints rather than as a working point. By doing so, the most promising constraints will be inserted one at a time. Unlike the Simplex method, this is possible for the Attic method since it is not forced to move only on vertices (see Figure 10.2). 

Table 10.1 shows the maximum number of iterations with respect to the problem dimensions that should be required by the Simplex method and by the Attic method to solve the linear programming problems with these features (e.g., the Klee-Minty problem). 

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. 

Buzzi-Ferraris, G. (2011b) Linear programming with the attic method. Industrial ej Engineering Chemistry Research, 50, 4858-4878. 

Vol. V Buzzi-Ferraris and Manenti, The Attic Method in Mixed Integer and Linear Programming for the Chemical Engineer Solving Numerical Problems, in progress. 

Chapter 10 introduces the Attic method for linear programming, and to which we dedicate a complete volume (Vol. 5 - Buzzi-Ferraris and Manenti, in progress) because of the huge significance of the method. The Attic method is compared to the existing Simplex and Interior Point methods, emphasizing its efficiency and robustness. 

Lattice Poisson-Boltzmann Method, Analysis of Electroosmotic attice Poisson-Boltzmann Method, Analysis of Bootroosmotic Microfludics, Figure 2 LPM results compared with the linearization Figure 3 Effects of the pseudo speed of sound values... 

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