Penalty Formulation

A constrained minimization problem in standard format

min f₀(x)

x in X

s.t. g_j(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)+r_pSum{max(0,g_j)²}, j=1,...,m

x in X

The measure of constraint violation, Sum{max(0,g_j)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 r_p = 1

The variable penalty factor is updated according to the following schedule:

r_p+1=cr_p

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.