Hierarchical Formulation

To handle constraints more efficiently, a constrained minimization problem in standard format

min f₀(x)

x in X

s.t. g_j(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. g_j(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{g_j(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.