A Multi-Objective Constraint-Handling Method with PSO Algorithm for Constrained Engineering Optimization Problems Lily D LI1, Xiaodong Li2, Xinghuo Yu2
-----------------------------------------------------------------------------------------------------------------
1
Central Queensland University, Australia 2 RMIT University, Australia IEEE Congress on Evolutionary Computation 2008
1
Outline
Background Problem formulation The proposed approach
PSO algorithm Constraint handling
Simulation Results Conclusion IEEE Congress on Evolutionary Computation 2008
2
Background
Real world optimization problems are constrained Constraint-Handling is a key issue Evolutionary algorithms for optimization successful in unconstrained optimization problems constraint-handling remain problematic because their original versions have no mechanism to incorporate constraints into fitness functions IEEE Congress on Evolutionary Computation 2008
3
Background (cont.)
Constraint-Handling via Genetic Algorithm has been studied Constraint-Handling via Particle Swarm Optimization Algorithm has few attempts
IEEE Congress on Evolutionary Computation 2008
4
Problem formulation
Constrained single objective optimization problem minimize f (X) subject to gi (X) ≤ 0,
⎫ ⎬ i = 1,2,....m.⎭
(1)
where X=[x1,x2,…xn] , with each xi (i=1,2,…n) is bounded by lower and upper limits Li ≤ xi ≤ Ui m is the total number of constraints
IEEE Congress on Evolutionary Computation 2008
5
Problem formulation (cont.)
Multi-objective constraint-handling method
Treat constraints as objectives
Transform single objective constrained problems into bi-objective unconstrained problems m in im iz e w h e re
F ( X )= ( f ( X ), Φ ( X )) Φ (X ) =
m
∑
i =1
⎫ ⎪ ⎬ m a x ( 0 , g i ( X )) ⎪ ⎭
(2 )
Φ is a total amount of constraint violations
IEEE Congress on Evolutionary Computation 2008
6
Problem formulation (cont.)
A general Multi-objective Optimization (MOO) problem
Find Pareto-front Apply high-level information to find the final solution
For this problem in Equation (2)
Constraints satisfaction (Φ = 0 or Φ ≤ ε) is a must Can be used as high-level information
IEEE Congress on Evolutionary Computation 2008
7
Problem formulation (cont.) The Pareto-front, feasible solutions and desired constrained minimum for a bi-objective constraint handling optimization problem
IEEE Congress on Evolutionary Computation 2008
8
Problem formulation (cont.)
If a predefined target is given The problem can be transferred into minimize goal
F(X)=(f (X) - t, Φ(X)) ⎫ ⎬ Φ(X) ≤ ε and f (X) - t ≤ Δ ⎭
(3)
t: predefined target ε: minimum error allowed to constraints Δ: minimum error allowed to target IEEE Congress on Evolutionary Computation 2008
9
Problem formulation (cont.)
The solutions area for a bi-objective constraint handling optimization problem
IEEE Congress on Evolutionary Computation 2008
10
Proposed approach
Integrate Multi-objective constraint-handling method into PSO algorithm Adopt domination concept from Multiobjective optimization in deciding particles behaviours
IEEE Congress on Evolutionary Computation 2008
11
PSO algorithm
stochastic method metaphor of social behaviour of flocks of birds and schools of fish link to evolutionary computation no “survival of the fittest”
IEEE Congress on Evolutionary Computation 2008
12
PSO algorithm
Formula v id
(t +1)
v id
(t +1)
v id
(t +1)
= χ [ w v id
)⎫ ⎪ + c 2 r 2 i d ( t ) ( l B e s t i d ( t ) − x i d ( t ) ) ] ⎪⎪ ⎬ i f v id ( t +1) > V m ax = V m ax , ⎪ ⎪ i f v id ( t +1) < − V m ax = − V m ax , ⎪⎭ (t )
+ c 1 r1id
(t )
( p B e s t id
(t )
− x id
(t )
(a) (b)
xid (t +1) = xid (t ) + vid (t +1)
vidt+1: velocity for particle i, dimension d, generation t+1 xidt+1: position for particle i, dimension d, generation t+1
IEEE Congress on Evolutionary Computation 2008
13
Constraint Handling
Multi-objective Constraint Handling Adopt domination concept Definition: A solution X(1) is said to dominate the other solution X(2), if both conditions 1 and 2 are true: The solution X(1) is no worse than X(2) in all objectives, (for all j =1,2,…,m). The solution X(1) is strictly better than X(2) in at least one objective, for at least one j ∈ {1, 2 , ..., m }
Second objective Φ has high priority
Final results must be selected from Φ = 0 or Φ ≤ ε
IEEE Congress on Evolutionary Computation 2008
14
Selection rules and Diversity Control
Selection rules
Non-dominated particles are better than dominated ones Particles with lower Φ is better than those with higher Φ (means closer to the feasible area)
Perturbation
minor probability p disturb few particles by regenerating IEEE Congress on Evolutionary Computation 2008
15
Modified PSO algorithm 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
GlobalF = POSITIVE_INFINITY; P0 = URand (Li, Ui) V0 = 0 F0 = Fitness_F (P0) Φ0 = Fitness_Φ(P0) pBest0 = P0 For i = 0 To n lBesti = LocalBest (Pi-1, Pi, Pi+1) End for Do For i = 0 To n Vi+1 = Speed ( Pi,,Vi, pBesti, lBesti) Pi+1 = Pi+ Vi+1 r = URand (0,1) If (r ≤ p) TempPi+1 = Rand (Li, Ui)
(Selection rules)
(Equation (a)) (Equation (b))
IEEE Congress on Evolutionary Computation 2008
16
Modified PSO algorithm (cont.) 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.
If (TempPi+1 isBetterThan Pi+1) Pi+1 = TempPi+1 Fi+1 = Fitness_F(Pi+1) Φi+1 = Fitness_Φ(Pi+1) If (Pi+1 isBetterThan pBesti ) pBesti = Pi+1 If (Φi+1 ≤ ε) If (Fi+1 < GlobalF) GlobalF = Fi+1 End For For i = 0 To n lBesti = LocalBest (Pi-1, Pi, Pi+1) End for If (GlobalF ≤ Δ ) Output GlobalF and Pi+1 Stop End Do
(Selection rules)
(Selection rules)
(Selection rules)
IEEE Congress on Evolutionary Computation 2008
17
Features of the modified PSO algorithm
Whenever calculating fitness, both objectives F=f(X)-t and Φ need to be evaluated;
If a particle’s new location is better than its best past location, the pBest is updated (decided by selection rules);
IEEE Congress on Evolutionary Computation 2008
18
Features of the modified PSO algorithm (cont.)
A particle’s best neighboring particle is determined by the two steps:
Find all the non-dominated particles in the neighborhood; If there is only one non-dominated particle in the neighborhood, select it as lBest; otherwise, select one with the lowest Φ as lBest (the lower Φ means closer to the feasible region).
IEEE Congress on Evolutionary Computation 2008
19
Features of the modified PSO algorithm (cont.)
A minor perturbation with the probability of p is introduced after the calculating the next particle position. If perturbed particle is better than the particle before perturbation, replace the particle with the perturbed one. After each iteration, check whether the goal is obtained; if obtained, the iteration ends. IEEE Congress on Evolutionary Computation 2008
20
Simulation
Three well-known engineering design problems
E01: Welded beam design problem (This problem has little difference from the original one proposed by Reklaitis et al in 1983 which has 5 constraints. For some reason, many researchers have studied the one has 7 constraints, we include this version for comparison purpose)
E02: Pressure vessel design problem E03: Spring design problem
IEEE Congress on Evolutionary Computation 2008
21
E01: Welded beam design problem Minimize f ( X ) = 1.10471x12 x2 + 0.04811x3 x4 (14.0 + x2 )
Subject to g1(X) =τ(X) −τmax ≤ 0 g2(X) =σ(X) −σmax ≤ 0 g3(X) = x1 − x4 ≤ 0 g4(X) = 0.10471x12 +0.04811x3x4(14+ x2) −5≤0 g5(X) =0.125− x1 ≤0 g6(X) =δ(X) −δmax ≤0 g7(X) = P−Pc (X) ≤0
IEEE Congress on Evolutionary Computation 2008
22
E02: Pressure vessel design problem
Minimize
f ( X ) = 0.6224 x1 x3 x4 + 1.7781x2 x32 + 3.1661x12 x4 + 19.84 x12 x3
Subject to
g 1 ( X ) = 0.0193 x 3 − x1 ≤ 0 g 2 ( X ) = 0.00954 x 3 − x 2 ≤ 0 g 3 ( X ) = 1296000 − π x 3 2 x 4 −
4 π x33 ≤ 0 3
g 4 ( X ) = x 4 − 240 ≤ 0
IEEE Congress on Evolutionary Computation 2008
23
E03: Spring design problem
Minimize Subject to
f ( X ) = ( x 3 + 2) x 2 x1 2 g1 ( X ) = 1 − g2 ( X ) =
4 x 2 2 − x1 x 2 1 + −1 ≤ 0 3 4 12566( x 2 x1 − x1 ) 5108 x1 2
g3 ( X ) = 1 − g4 ( X ) =
x 2 3 x3 ≤0 71785 x1 4
140.45 x1 ≤0 x 2 2 x3
x 2 + x1 −1 ≤ 0 1.5 IEEE Congress on Evolutionary Computation 2008
24
Parameters
Neighbourhood topology: ring (circular) w = 0 (Empirically derived) c1=c2=2 χ=0.63 Vmax= 0.5*(decision variable range) p = 0.1% number of particles is 100 the maximum iteration is set to 10,000 ε = 1.0E-9 Δ = 1.0E-04 (E01 and E03), 1.0E-01 (E02), because E02 has large magnitude than E01 and E03. 30 independent runs IEEE Congress on Evolutionary Computation 2008
25
Result for E01 - Design variables Comments: The best-known: 1.724852 Our best result: 1.724852321 Mean: 1.724861948 Standard dev.: 2.05462E-05 Search quality: good Consistency: good
IEEE Congress on Evolutionary Computation 2008
26
Result for E01 - Algorithm performance Comments: Best solution found: Better than other three Mean result found: Better than other three Standard derivation: Better than C&M Slightly worse than CPSO and HES-PSO but acceptable
IEEE Congress on Evolutionary Computation 2008
27
Result for E02 - Design variables Comments: The best-known: 6059.94634 Our best result: 5971.4003 Mean: 6049.1590 Standard dev.: 22.841537 Search quality: good Consistency: good
IEEE Congress on Evolutionary Computation 2008
28
Result for E02 - Algorithm performance Comments: Best solution found: Better than other three Mean result found: Not as good as other three Standard derivation: Not as good as other three By looking up Table III, it seems our approach needs more iterations to achieve consistent results.
IEEE Congress on Evolutionary Computation 2008
29
Result for E03 - Design variables Comments: The best-known: 0.012665 Our best result: 0.012665236 Mean: 0.012714543 Standard dev.: 6.28E-05 Search quality: good Consistency: good
IEEE Congress on Evolutionary Computation 2008
30
Result for E03 - Algorithm performance Comments: Best solution found: Better than other three Mean result found: Similar to other three Standard derivation: Similar other three
IEEE Congress on Evolutionary Computation 2008
31
Discussion • Performance
Comments: Depend on Φ and Δ larger Φ and Δ will make the search easy We use smaller Φ and Δ ε = 1.0E-9 Δ = 1.0E04, 1.0E01 and 1.0E04 Efficient Rational cost
IEEE Congress on Evolutionary Computation 2008
32
Conclusion
Multi-objective constraint handling method Adopt domination concept Selection rule Perturbation for diversity No problem-dependent parameters Rational computation cost Simulation to 3 engineering design problems demonstrated: capability, efficiency Methodology can be applied to wider applications IEEE Congress on Evolutionary Computation 2008
33
