1. Chapter 7 Handling Constraints

2. NonLinear Programming Problem
NLP with linear constraints:
Optimize
Domain constraints:
Equalities:
Inequalities:

3. GENOCOP I
Original GENOCOP (GEnetic algorithm, for Numerical Optimization for COnstrained Problems):
With linear constrain
An elimination of the equalities (convex)
Special “genetic” operators

4. Example Optimize a function of six variables:
subject to the following constraints
Express four variables as functions of the remaining two:

5. Reduce the original problem to the optimization problem of a function of two variables and :
Subject to the following constraints (inequalities only):
These inequalities can be further reduced to:

6. Elimination of Equalities
Equality constraint set:
Split A:
New set of inequalities (after removal ):
Split C:

7. Final set of constraints:
original domain constraints:
new inequalities:
original inequalities (after removal of variables):

8. Example (1) Optimize a function of six variables:
subject to the following constraints
Domain constraints:
Equalities:
Inequalities:

9. Example (2)
Transportation problem:

10. Initialization process
Representation
floating point representation  
Initialization process
A subset of potential solutions -- the space of the whole feasible region (randomly)
The remaining subset -- the boundary of the solution space.
Genetic operators
dynamic
non-uniform.

11. Mutation
Uniform mutation
Boundary mutation
Non-uniform mutation

12. Crossover
Arithmetical crossover
Simple crossover
Heuristic crossover

13. GENOCOP II With non-linear constrain
Distinguish between linear and nonlinear constraints
A single starting point
Quadratic penalty function
Iterative execution of GENOCOP

14. Algorithm
Procedure GENOCOP II
begin
split the set of constraints C into
select a starting point ( need not be feasible.)
set the set of active constraints, A to (V: violated constraints at point )
set penalty

15. while (not termination-condition) do
begin
execute GENOCOP I for the function
with linear constraints L and the starting point
save the best individual :
update A:
decrease penalty r:
(where ;
end

16. Example Minimize s.t. Iteration Number 1 2 3 4 The best point (0,0)1 2 3 4
The best
point
(0,0)
(3,4)
(2.06, 3.98)
(2.3298, )
(2.3295, )
Active
Constraints
none c2
c1 , c2

17. Other Techniques
Homaifar
Joines and Houck

18. Schoenauer and Xanthakis
Start with a random population of individuals (feasible or infeasible)
Set ( j is a constraint counter)
Evolve this population with , until a given percentage of population (flip threshold ) is feasible for this constraint
Set
The current population is the starting point for the next phase of the evolution, where
If , repeat the last two steps, otherwise optimize the objective function ,

19. Powell and Skolnick

20. Bean and Hadj-Alouane

21. GENOCOP III Two separate populations Repair:
(search point): satisfy linear constraints
(reference point): satisfy all constraints
Repair:
Feasible points: (reference point )
Infeasible search points: (search point )
( : better reference points)
( is feasible)
( :probability of replacement)
(if is better than )
( :probability of replacement)

23. Extend GENOCOP III
Nonlinear equations