Penalty Function

The significant step after getting solution in all generations is to check the feasibility of penalize and solution the violating constraints. Usually, there exist mostly four types of penalty functions name is: constant, dynamic, static and adaptive. Static penalty function penalizes infeasible solution via imposing a constant to each constraint violated. Generally, this can be represented as like:

Described an optimization problem

                                min f ( x) s . t . x ∈ A;x ∈ B      . . . Eqn1

 Here x is a vector of decision variables, ' x ∈ A ' are relatively simple to satisfy the constraints

' x ∈ B ' are relatively not easy to satisfy. The problem can be formulated as like:

                               min f ( x) + p (d ( x, B)) s . t . x ∈ A    . . . Eqn2

 Here d (x, B) is a metric function explaining the distance of vector 'x' from region B, and p (d) is a penalty function as p (0) = 0. It is an exterior penalty function such  as if p (d) grows quickly sufficient outside of B then optimal solution of Eq.1 will be optimal also for Eq. 2.