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

min f_{0}(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 f_{0}(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: |

_{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. |

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