Bound-Constrained Formulation for Lagrangian Penalty Function

Bound-Constrained Formulation for Lagrangian Penalty Function 

As already emphasized, it is crucial to insert the bounds on each variable in the constrained minimization problems since the search region is reduced, and special features of said bounds can be exploited. 

As demonstrated in the previous section, if any inequality constraints were present, it might be sufficient to introduce a new slack variable for each of them and a lower bound for normegativity. The upper bound will have an adequately large value. Thus, the presence of inequality constraints enters into the formulation case of the problem (12.35) and (12.36). 

The problem (12.35) and (12.36) is now transformed into the minimization of the augmented Lagrangian function  

This problem can be solved by using one of the algorithms adopted to handle minimization problems with bound constraints only after having assigned an initial guess to the parameters Aj and vj (see Chapter 13). When a satisfactory minimum is found, the Lagrange multipliers Ij are updated using (12.29) and the values of the parameters vj are increased. 

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