# 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:  rp+1=crp 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. 