Differential Evolution Optimization Lecture
DIFFERENTIAL EVOLUTION
DE
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Differential Evolution Optimization Lecture
Outline
Outline
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Rationale
Hystorical notes
DE basics
Implementation (Matlab code)
Improvement strategies
Constraint handling
Conclusions
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Differential Evolution Optimization Lecture
Rationale
Rationale
■  Why are stochastic methods needed?
•
•
•
•
•
•
Mixed real/integer parameter problems (e.g. number of poles/slots)
Constraints (explicit/implicit)
Noisy objective functions (due to numerically evaluated objectives)
Multiminima
Multiobjective
…..
■  Drawback: Very expensive
N. evaluations
■  Parallelism ?
500.000
■  Target: 1 day run
50.000
5.000
T [s]
Solver
0.15
Simple analytical
1.5
Complex analytical
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2d FEM
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Differential Evolution Optimization Lecture
Hystorical notes
Hystorical notes
■  Differential Evolution: what and when
•  R. Storn and K. Price, “Differential
Evolution - A Simple and Efficient
Adaptive Scheme for Global
Optimization over Continuous
Spaces,” Tech. Report, International
Computer Science Institute
(Berkeley), 1995.
•  R. Storn and K. Price, “Differential
Evolution – A Simple and Efficient
Heuristic for global Optimization
over Continuous Spaces,” Journal
of Global Optimization, vol. 11, Dec.
1997, pp. 341-359.
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Differential Evolution Optimization Lecture
DE Basics
Algorithm
DE Basics
■  Algorithm: Typical evolutionary scheme
■  Mutation: Expands the search space
■  Recombination: Reuses previously successful individuals
■  Selection (Explicit): Mimics survival-of-the-fittest
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Differential Evolution Optimization Lecture
DE Basics
Mutation
DE Basics
■  Mutation: add difference vector(s) to a base individual in
order to explore the search space
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■
■
■
■
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■
DE/rand/1
DE/best/1
DE/rand to best/1
DE/curr. to best/1
DE/rand/2
DE/best/2
…
vi
vi
vi
vi
= xr1 + F1 (xr2 − xr3 )
= xbest + F1 (xr2 − xr3 )
= xr1 + F1 (xr2 − xr3 ) + F2 (xbest − xr1 )
= xi + F1 (xr2 − xr3 ) + F2 (xbest − xi )
vi = xr1 + F1 (xr2 − xr3 + xr4 − xr5 )
vi = xbest + F1 (xr2 − xr3 + xr4 − xr5 )
donor
multiobjective ???
mutation factor
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Differential Evolution Optimization Lecture
DE Basics
Recombination
DE Basics
■  Recombination: mix successful solutions from the
previous generation with current donors
ui,j =
trial
�
vi,j
xi,j
target
if randi,j ≤ CR or j = irand
else
donor
crossover ratio
§  Scheme ensures that at least one DOF is changed
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Differential Evolution Optimization Lecture
DE Basics
Selection
DE Basics
■  Selection: Greedy scheme is key for fast convergence of DE
xk+1
=
i
�
uki
k
xi
if f (uki ) < f (xki )
else
multiobjective???
§  The power of DE was shown at the First International Contest on Evolutionary
Optimization in May 1996 (IEEE International Conference on Evolutionary
Computation). DE was the best general purpose algorithm.
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Differential Evolution Optimization Lecture
Single objective DE
Implementation
Matlab code
1 clear all; close all; clc
2 %Function to be minimized
3 D=2;
4 objf=inline(’4*x1^2é2.1*x1^4+(x1^6)/3+x1*x2é4*x2^2+4*x2^4’,’x1’,’x2’);
5 objf=vectorize(objf);
6 %Initialization of DE parameters
7 N=20; %population size (total function evaluations will be itmax*N, must be
>=5)
8 itmax=30;
9 F=0.8; CR=0.5; %mutation and crossover ratio
10 %Problem bounds
11 a(1:N,1)=é1.9; b(1:N,1)=1.9; %bounds on variable x1
12 a(1:N,2)=é1.1; b(1:N,2)=1.1; %bounds on variable x2
13 d=(béa);
14 basemat=repmat(int16(linspace(1,N,N)),N,1); %used later
15 basej=repmat(int16(linspace(1,D,D)),N,1); %used later
16 %Random initialization of positions
17 x=a+d.*rand(N,D);
18 %Evaluate objective for all particles
19 fx=objf(x(:,1),x(:,2));
20 %Find best
21 [fxbest,ixbest]=min(fx);
22 xbest=x(ixbest,1:D);
23 %Iterate
24 for it=1:itmax;
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permat=bsxfun(@(x,y) x(randperm(y(1))),basemat’,N(ones(N,1)))’;
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%Generate donors by mutation
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v(1:N,1:D)=repmat(xbest,N,1)+F*(x(permat(1:N,1),1:D)éx(permat(1:N,2),1:
D));
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%Perform recombination
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r=repmat(randi([1 D],N,1),1,D);
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muv = ((rand(N,D)<CR) + (basej==r)) ~= 0;
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mux = 1émuv;
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u(1:N,1:D)=x(1:N,1:D).*mux(1:N,1:D)+v(1:N,1:D).*muv(1:N,1:D);
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%Greedy selection
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fu=objf(u(:,1),u(:,2));
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idx=fu<fx;
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fx(idx)=fu(idx);
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x(idx,1:D)=u(idx,1:D);
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%Find best
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[fxbest,ixbest]=min(fx);
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xbest=x(ixbest,1:D);
41 end %end loop on iterations
42 [xbest,fxbest]
Please note that this code
actually works !
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Differential Evolution Optimization Lecture
Single objective DE
Influence of parameters
DE
■  So far so good, but what about:
■  Mutation strategy ?
■  Mutation factor ?
■  Crossover ratio ?
■  What impact do they have ?
■  Can general rules be found or is fiddling necessary?
■  Can other ideas be incorporated ?
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Differential Evolution Optimization Lecture
Single objective DE
Influence of parameters (mutation strategy)
DE/rand/1/bin
f (x) =
N
−1
�
i=1
■  vi = xr1 + F1 (xr2 − xr3 )
[(1 − xi )2 + 100(xi+1 − x2i )]
Low-dimensional
5000 eval.
High-dimensional
■  Curse of dimensionality (1M eval. to reach convergence)
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Differential Evolution Optimization Lecture
Single objective DE
Influence of parameters (mutation strategy)
DE/rand/2/bin
■  Add two difference vectors instead of one
vi = xr1 + F1 (xr2 − xr3 + xr4 − xr5 )
■  Not so good as DE/rand/1/bin
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Differential Evolution Optimization Lecture
Single objective DE
Influence of parameters (mutation strategy)
DE/best/1/bin
■  Always start from best
vi = xbest + F1 (xr2 − xr3 )
■  Much faster convergence than DE/rand/1/bin
■  Stagnation?
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Differential Evolution Optimization Lecture
Single objective DE
Influence of parameters (mutation strategy)
DE/best/2/bin
■  Always start from best but use two difference vectors
vi = xbest + F1 (xr2 − xr3 + xr4 − xr5 )
■  Similar convergence to DE/rand/1/bin in 2D
■  Rather poor performance in 30D
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Differential Evolution Optimization Lecture
Single objective DE
Influence of parameters (mutation strategy)
DE/rand-to-best/1/bin
■  Include movement to best (analogy with PSO)
vi = xr1 + F1 (xr2 − xr3 ) + F2 (xbest − xr1 )
■  Faster convergence than DE/rand/1/bin
■  Stagnation?
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Differential Evolution Optimization Lecture
Single objective DE
Influence of parameters (mutation strategy)
Comparison
■  And the winner is….. DE/current-to-best/1/bin
■  30D problem very tough
■  Some better ideas are needed!
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Differential Evolution Optimization Lecture
Improvement strategies
Improvement strategies (more aggressive behaviour)
Aggressive strategy (asynchronous)
■  DE/rand/1:Use good solutions immediately
■  Good improvement but only in 2D
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Differential Evolution Optimization Lecture
Improvement strategies
Improvement strategies (more aggressive behaviour)
Aggressive strategy (asynchronous)
■  DE/current-to-best/1:Use good solutions immediately
■  Good improvement but only in 2D
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Differential Evolution Optimization Lecture
Improvement strategies
Improvement strategies (Changing F)
Changing F
■  Improve solution by changing mutation factor
■  Faster and/or deeper convergence
■  Adaptivity? YES
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Differential Evolution Optimization Lecture
Improvement strategies
Improvement strategies (Population size)
Population size
■  Smaller populations work well
■  Faster and/or deeper convergence
■  Adaptivity? YES (kill old, add if stagnating)
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Differential Evolution Optimization Lecture
Improvement strategies
Improvement strategies (Elitism)
Elitism (NEW)
■  Compare trial not only with target but also with worst
■  Faster and/or deeper convergence
■  Useful
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Differential Evolution Optimization Lecture
Improvement strategies
Improvement strategies (Coevolution)
Coevolution
■  Several populations, each working on a subset of the
degrees of freedom
Population 1:
x=(x1,2)
After iteration:
xbest=(1.2,2), f=9.9
Current best:
x=(1,2), f=10.0
Hopefully:
x=( 1.2, 2.5 ), f=9.0
Population 2:
x=(1,x2)
After iteration:
xbest=(1,2.5), f=9.5
■  Recent work suggests that coevolution may exhibit some
“free lunches”… for us a “free sandwich” is enough…
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Differential Evolution Optimization Lecture
Improvement strategies
Improvement strategies (Coevolution)
Coevolution
■  Several populations, each working on a subset of the
degrees of freedom
■  Faster convergence
■  Useful
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Differential Evolution Optimization Lecture
Improvement strategies
Final comparison
§  Not mentioned here:
■  Summing up
§  Comparison with nearest target
§  F from Gaussian or Cauchy distrib.
§  …
■  The combined use of improvement techniques
(agggressive+elitist+coevolutionary) allows to solve highly
dimensional 2d FEM problems in 1 day!
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Differential Evolution Optimization Lecture
Conclusions
Conclusions
■  Hybrid DE can efficiently solve problems with:
•
•
•
•
Mixed real/integer parameters
Constraints
Noisy objective functions
Multiminima
■  Parallel computing allows (simple) stochastic 3d FEM
optimization
N. generations
N
N. eval.
T single eval. [s]
T eval. [s]
Solver
25.000
20
500.000
3
idem
Very complex analytical
2.500
20
50.000
30
idem
Average 2d FEM
250
20
5.000
300 = 5 min
idem
Simple 3d FEM
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Differential Evolution Optimization Lecture
Additional material
■  If time permits…
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Differential Evolution Optimization Lecture
DE vs. PSO
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Differential Evolution Optimization Lecture
Constraint handling
Existing approaches and proposal
Constraint handling
■  Existing approaches:
•  Lagrange multipliers (continuous only)
•  Penalty (value?)
•  …
■  Elegant solution:
•  Domination in a single-objective context
•  Incorporate in selection operator:
•  If both candidate and target are feasible choose best
•  If candidate/target is feasible and target/candidate unfeasible choose
feasible
•  If both candidate and target are unfeasible choose the least
unfeasible
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Differential Evolution Optimization Lecture
Multiobjective DE
Proposed method
Multiobjective DE
■  Proposed method:
•  DE/current-to-best/aggressive/elitist/multipopulation with following
modifications:
•  All targets and all trial individuals are non-dominated,least-crowded
sorted in the selection phase (like NSGA-II)
•  Best half evolve to next generation
•  “best” individual is randomly selected from level-1 front
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Differential Evolution Optimization Lecture
Multiobjective DE
Analytical benchmark
Multiobjective DE
■  Analytical benchmark (KUR100)
■  Proposed multiobjective DE dominates NSGA-II and
SPEA2 (Strength Pareto Evolutionary Algorithm)
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Differential Evolution Optimization Lecture
Validation
Motor benchmark
Constrained single-objective DE
■  Motor benchmark
!!#
!"#
!"$
!$!
Method
Avg. torque [Nm] Torq. ripple [Nm] Torque ripple %
Reference
3.99
1.24
31%
TRIBES (PSO)
3.82
0.46
12%
DE
3.95
0.43
11%
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Differential Evolution Optimization Lecture
Validation
Motor benchmark
Constrained multi-objective DE
■  Motor benchmark
■  Maximize average torque / Minimize torque ripple
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