Hierarchical Formulation

To handle constraints more efficiently, a constrained minimization problem in standard format 
min f0(x)
x in X
s.t. gj(x) < 0, j=1,...,m
is modified using a hierarchy of objective functions.
It is possible that the initial point is not feasible. In that case, a phase I optimization problem 
min g(x)
x in X
is solved until a feasible point is found:
After the first feasible point is found the optimization proceeds in the following manner:
min f0(x) ( primary objective )
min g(x) (secondary objective)
x in X
s.t. gj(x) < 0, j=1,...,m
where the primary objective function is the original one, and the secondary objective function is defined as follows:
g(x)= max{gj(x)}, j=1,...,m
In the case when two designs give equal values of the primary objective the secondary objective function is minimized which results in a design "deeper" into the interior of the feasible design space.

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