Selection of the Search Direction

Let us now discuss the crucial problem of the selection of the vector h. When a component of the vector h in the system (10.17) is null, its corresponding constraint is still satisfied. 

If h 0 is imposed for an inequality constraint, it is made passive. It is not possible to assign h 0 for an active inequality constraint since, in that instance, the search direction would violate the constraint 

Additional artificial constraints do not impose any limitation on the choice of the corresponding value of ha- 

The solution d of the system (10.17) depends on the choice of the vector h. That direction leads to a function decrease if it results in 

It is worth noting that if we select the values he = 0 for the equality constraints ha = Xa, hj = Xj, and h = 0, the search direction is the one that maximizes the objective function decrease, and simultaneously satisfies all the active equality and inequality constraints. 

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