Penalty Formulation

A constrained minimization problem in standard format 
min f0(x)
x in X
s.t. gj(x), j=1,...,m
can be solved using an exterior penalty formulation in which a penalty is imposed for each constraint violation. 
The resulting penalty formulation is as follows:
min F(x)=f0(x)+rpSum{max(0,gj)2}, j=1,...,m
x in X
The measure of constraint violation, Sum{max(0,gj)2}, j=1,...,m has a value equal to zero in the feasible region, and a positive value when any constraint is violated. Here, the variable penalty approach is adopted to iteratively increase the penalty factor. 

This program allows a user to specify an initial value for the penalty factor, or choose a default value:

default initial penalty factor rp = 1 
The variable penalty factor is updated according to the following schedule: 


where the updating constant c, and the number of iterations between each penalty iteration may be specified by the user, or set to default values:

default updating constant c= 1.61803 (golden section ratio)

default number of iterations between two consecutive penalty iterations = 100.

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