a multi-objective constraint-handling method with pso algorithm for constrained engineering...

A Multi-Objective Constraint-Handling Method with PSO Algorithm for Constrained Engineering Optimization Problems

Lily D LI1, Xiaodong Li2, Xinghuo Yu2

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1 Central Queensland University, Australia2 RMIT University, Australia

1IEEE Congress on Evolutionary Computation 2008

OutlineBackgroundProblem formulationThe proposed approach

PSO algorithmConstraint handling

SimulationResultsConclusion


BackgroundReal world optimization problems are constrainedConstraint-Handling is a key issueEvolutionary algorithms for optimization

successful in unconstrained optimization problemsconstraint-handling remain problematic because their original versions have no mechanism to incorporate constraints into fitness functions


Background (cont.)Constraint-Handling via Genetic Algorithm has been studied Constraint-Handling via Particle Swarm Optimization Algorithm has few attempts


Problem formulationConstrained single objective optimization problem

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

minimize ( ) (1)

subject to ( ) 0, 1,2,.... .i

fg i m

⎫⎬≤ = ⎭

XX


Problem formulation (cont.)Multi-objective constraint-handling method

Treat constraints as objectives

Transform single objective constrained problems into bi-objective unconstrained problems

Φ is a total amount of constraint violations

1

m in im ize F ( )= ( ( ), ( )) (2 )

w h ere ( ) m ax (0 , ( )) m

ii

f

g=

Φ ⎫⎪⎬Φ = ⎪⎭

∑

X X X

X X


Problem formulation (cont.)A general Multi-objective Optimization (MOO) problem

Find Pareto-frontApply 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


Problem formulation (cont.)The Pareto-front, feasible solutions and desired constrained minimum for a

bi-objective constraint handling optimization problem


Problem formulation (cont.)If a predefined target is givenThe problem can be transferred into

t: predefined targetε: minimum error allowed to constraintsΔ: minimum error allowed to target

minimize F(X)=( ( ) - , (X)) (3)

goal (X) and ( ) - f t

f tεΦ ⎫

⎬Φ ≤ ≤Δ ⎭

XX


Problem formulation (cont.)The solutions area for a bi-objective constraint handling optimization problem


Proposed approachIntegrate Multi-objective constraint-handling method into PSO algorithmAdopt domination concept from Multi-objective optimization in deciding particles behaviours


PSO algorithmstochastic methodmetaphor of social behaviour of flocks of birds and schools of fishlink to evolutionary computationno “survival of the fittest”

IEEE Congress on Evolutionary Computation 2008

12

PSO algorithmFormula

(a)

(b)

vidt+1: velocity for particle i, dimension d, generation t+1

xidt+1: position for particle i, dimension d, generation t+1


1 1

2 2

( 1 ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( 1 ) ( 1 )m a x m a x

( 1 ) ( 1 )m a x m a x

[ + ( )

+ ( ) ]

,

,

i d i d i d i d i d

i d i d i d

i d i d

i d i d

t t t t t

t t t

t t

t t

v w v c r p B e s t x

c r l B e s t x

v V i f v V

v V i f v V

χ+

+ +

+ +

⎫= −⎪

− ⎪⎪⎬

= > ⎪⎪= − < − ⎪⎭

( 1) ( ) ( 1)id id id

t t tx x v+ += +

Constraint HandlingMulti-objective Constraint HandlingAdopt domination conceptDefinition: 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

Second objective Φ has high priorityFinal results must be selected from Φ = 0 or Φ ≤ ε


{1, 2 , ..., }j m∈

Selection rules and Diversity ControlSelection rules

Non-dominated particles are better than dominated onesParticles with lower Φ is better than those with higher Φ (means closer to the feasible area)

Perturbationminor probability pdisturb few particles by regenerating


Modified PSO algorithm


1. GlobalF = POSITIVE_INFINITY; 2. P0 = URand (Li, Ui)3. V0 = 04. F0 = Fitness_F (P0)5. Φ0 = Fitness_Φ(P0)6. pBest0 = P07. For i = 0 To n 8. lBesti = LocalBest (Pi-1, Pi, Pi+1) (Selection rules)9. End for10. Do 11. For i = 0 To n12. Vi+1 = Speed ( Pi,,Vi, pBesti, lBesti) (Equation (a)) 13. Pi+1 = Pi+ Vi+1 (Equation (b)) 14. r = URand (0,1)15. If (r ≤ p) 16. TempPi+1 = Rand (Li, Ui)

Modified PSO algorithm (cont.)


17. If (TempPi+1 isBetterThan Pi+1) (Selection rules)18. Pi+1 = TempPi+119. Fi+1 = Fitness_F(Pi+1)20. Φi+1 = Fitness_Φ(Pi+1)21. If (Pi+1 isBetterThan pBesti ) (Selection rules)22. pBesti = Pi+123. If (Φi+1 ≤ ε)24. If (Fi+1 < GlobalF) 25. GlobalF = Fi+126. End For27. For i = 0 To n28. lBesti = LocalBest (Pi-1, Pi, Pi+1) (Selection rules)29. End for30. If (GlobalF ≤ Δ ) 31. Output GlobalF and Pi+132. Stop33. End Do

Features of the modified PSO algorithm

Whenever calculating fitness, both objectivesF=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);


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).


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.


20

SimulationThree 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 problemE03: Spring design problem


E01: Welded beam design problem


22

Minimize2

1 2 3 4 2( ) 1.10471 0.04811 (14.0 )f X x x x x x= + +

Subject to

1 max

2 max

3 1 42

4 1 3 4 2

5 1

6 max

7

( ) ( ) 0( ) ( ) 0( ) 0

( ) 0.10471 0.04811 (14 ) 5 0( ) 0.125 0( ) ( ) 0( ) ( ) 0c

g X Xg X Xg X x x

g X x x x xg X xg X Xg X P P X

τ τσ σ

δ δ

= − ≤

= − ≤= − ≤

= + + − ≤

= − ≤

= − ≤= − ≤

E02: Pressure vessel design problem


23

Minimize 2 2 21 3 4 2 3 1 4 1 3( ) 0.6224 1.7781 3.1661 19.84f X x x x x x x x x x= + + +

Subject to 1 3 1

2 3 2

2 33 3 4 3

4 4

( ) 0.0193 0( ) 0.00954 0

4( ) 1296000 03

( ) 240 0

g X x xg X x x

g X x x x

g X x

π π

= − ≤= − ≤

= − − ≤

= − ≤

E03: Spring design problem


24

Minimize 23 2 1( ) ( 2)f X x x x= +

Subject to3

2 31 4

12

2 1 22 3 4 2

2 1 1 1

13 2

2 3

2 14

( ) 1 071785

4 1( ) 1 012566( ) 5108

140.45( ) 1 0

( ) 1 01.5

x xg Xx

x x xg Xx x x x

xg Xx x

x xg X

= − ≤

−= + − ≤

−

= − ≤

+= − ≤

ParametersNeighbourhood topology: ring (circular)w = 0 (Empirically derived)c1=c2=2χ=0.63 Vmax= 0.5*(decision variable range)p = 0.1% number of particles is 100the maximum iteration is set to 10,000ε = 1.0E-9Δ = 1.0E-04 (E01 and E03), 1.0E-01 (E02), because E02 has large magnitude than E01 and E03.30 independent runs


Result for E01- Design variables

Comments:The best-known: 1.724852Our best result: 1.724852321Mean: 1.724861948Standard dev.: 2.05462E-05Search quality: goodConsistency: good


26

Result for E01- Algorithm performance


Comments:Best solution found:

Better than other threeMean result found:

Better than other threeStandard derivation:

Better than C&MSlightly worse than CPSO and HES-PSO but acceptable



Comments:The best-known: 6059.94634 Our best result: 5971.4003 Mean: 6049.1590 Standard dev.: 22.841537 Search quality: goodConsistency: good





Not as good as other threeStandard derivation:

Not as good as other threeBy looking up Table III, it seems our approach needs more iterations to achieve consistent results.



Comments:The best-known: 0.012665 Our best result: 0.012665236 Mean: 0.012714543 Standard dev.: 6.28E-05 Search quality: goodConsistency: good





Similar to other threeStandard derivation:

Similar other three

Discussion• Performance


Comments:Depend on Φ and Δ

larger Φ and Δ will make the search easy

We use smaller Φ and Δε = 1.0E-9 Δ = 1.0E04, 1.0E01 and 1.0E04

EfficientRational cost

ConclusionMulti-objective constraint handling methodAdopt domination conceptSelection rulePerturbation for diversityNo problem-dependent parametersRational computation costSimulation to 3 engineering design problems demonstrated: capability, efficiencyMethodology can be applied to wider applications


a multi-objective constraint-handling method with pso algorithm for constrained engineering...

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