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Global Optimization Techniquesin Computational Electromagnetics
Zbyněk Raida
Dept. of Radio ElectronicsBrno University of TechnologyBrno, Czechia
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Outline
• What does the optimization mean
• Classification of optimization tasks- single-objective versus multi-objective- local versus global
• Genetic optimization vs. particle swarm one
• Local tuning of global solutions
• An example
Global optimization techniques …ITSS 2007, Pforzheim
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Optimizationdefinition
• Searching for such values of state variables to meet desired parameters as close as possible
ITSS 2007, Pforzheim
32.3
45.026.0
21.1
38.3 16.9
22.3
24.8
Global optimization techniques …
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Optimizationobjective function (1)
• Deviation of the actual parameters of the system from the desired ones
4
111 ,
nnfsF xx
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Optimizationobjective function (2)
0.5 1.0 1.5 2.0 2.5 3.0f [GHz]
computedmeasured
S11[dB]
-10
-15
-20
-25
-30
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More objectivespolarization purity (1)
ČÁP, A., RAIDA, Z., HERAS PALMERO, E., LAMADRID RUIZ, R. Multi-band planar antennas: a comparative study. Radioengineering, 2005, vol. 14, no. 4, p. 11–20.
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More objectivespolarization purity (2)
RAIDA, Z., HERAS PALMERO, E., LAMADRID RUIZ, R. Four-band patch antenna with U-shaped notches. In Proc. of the16th international Conference on Microwaves, Radar and Wireless Communications MIKON 2006. Krakow (Poland), 2006, pp. 111–114.
ITSS 2007, Pforzheim
a) b) c)
d)
Global optimization techniques …
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More objectivesdirectivity patterns (1)
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More objectivesdirectivity patterns (2)
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More objectivesmulti-objective formulation
4
1
0,,n
nhormaxD fEEF xx
4
111 ,
nnS fsF xx
4
1
90
90
,,n m
mnvertP fEF xx
ITSS 2007, Pforzheim
F
F
2
F
13
S
P
D
Global optimization techniques …
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Multi-objective optimizationtwo approaches min F ( x)S
min F ( x)D
min F ( x)P
multi-objectiveoptimizer
trade-offsolutions
higher-levelinformation
choose onesolution
min F ( x)S
min F ( x)D
min F ( x)P
single-object.optimizer
higher-levelinformation
one optimumsolution
estim. relative
S
importance
D
vector
P[w , w , w ]
single-object.optimization
F = w F + w F + w F
S
D P
S
D P
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Searching for a minimumglobal versus local methods
ITSS 2007, Pforzheim
f(x)
x
startingpoint
global local
Global optimization techniques …
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Global methodsgenetic algorithms (1)
mm00.9mm,00.1Amm050.0mm,001.0B
2.2,0.2,6.1,0.1r mm5.1mm,0.1h
GHz30f
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Global methodsgenetic algorithms (2)
initial populationquality evaluation
selection
ITSS 2007, Pforzheim
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Global methodsgenetic algorithms (3)
crossover
mutation
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function x = main( G, I, pc, pm)
% x(1)= A, x(2)= B, x(3)= h, x(4)= eps
load dip_616; % loading neural model
Rd = 200.0; % desired input resistanceXd = 0.0; % desired input reactancebit = [ 8 8 1 2]; % bits per A, B, h, epsgeb = norm( bit, 1) + 1; % bits in chromosome
gen = round( rand( I, geb-1)); % 1st generationfor g=1:G X = decode( I, bit, gen); % chromosome to A,B,h,eps Z = Tmax * sim( net, X'); % analysis gen(:,geb) = ((Rd-Z(1,:)).^2 + (Xd-Z(2,:)).^2; e(g) = min( gen( :,geb)); % minimum error [val,ind] = min( gen( :,geb)); x = X( ind, :); % best parameters gen = decim( gen, pc, pm, I, geb);end
plot( e);
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Global methodsgenetic algorithms (5)
0 5 10 15 20 25 30 35 40 iter.0
500
1000
1500
2000
2500
cost[ ]2
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Global methodsgenetic algorithms (6)
cost [2]
A [mm]
B [mm]
h [mm]
eps [ – ]
Rin []
Xin []
19 836 7.469 0.008 1.0 2.2 61.0 22.7
20 650 3.875 0.035 1.5 2.0 67.2 –54.9
402 5.156 0.026 1.5 1.0 183.3 –11.1
99 5.188 0.032 1.0 1.0 190.8 3.8
50 generations, 20 individuals, 90 % crossover, 10 % mutation, population decimation
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Global methodsparticle swarm optimization (1)
ROBINSON, J., RAHMAT-SAMII, Y. Particle swarm optimization in electromagnetics. IEEE Transactions on Antennas and Propagation. 2004, vol. 52, no. 2, p. 397–407.
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Global methodsPSO (2)
Tnnnnn hBA x
nnnnnn rcrcw xgxpvv 2211
nnn t vxx x
x
1
2
p2
g2
2
1
p11
ITSS 2007, Pforzheim
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Global methodsparticle swarm optimization (3)
absorbing reflecting invisible
ITSS 2007, Pforzheim
x
x
2
1
x
x
2
1
x
x
2
1
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function out = main( G, I)
% x(1)= A, x(2)= B, x(3)= h, x(4)= eps
load dip_616; % loading antenna model
Rd = 200; % required input resistanceXd = 0; % required input reactance
dt = 0.1; % time stepc1 = 1.49; % personal scaling factorc2 = 1.49; % global scaling factor
x = zeros( I, 5); % agents’ positionp = zeros( I, 5); % personal best
for n=1:I x(n,1) = 1.000 + 8.000*rand(); p(n,1) = x(n,1); x(n,2) = 0.001 + 0.049*rand(); p(n,2) = x(n,2); x(n,3) = 1.0 + 0.5 * rand(); p(n,3) = x(n,3); x(n,4) = 1.0 + 1.2 * rand(); p(n,4) = x(n,4); p(n,5) = 1e+6;end
v = rand( I, 4); % agent velocityg = zeros( 1, 4); % global beste = zeros( G+1, 1); e(1) = 1e+6;
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for m=1:G % +++ MAIN ITERATION LOOP +++
w = 0.5*(G-m)/G + 0.4; % inertial weight Z = Tmax * sim( net, x(:,1:4)'); % impedance of agents x(:,5) = ((Rd-Z(1,:)).^2 + (Xd-Z(2,:)).^2 [e(m+1),ind] = min( x( :,5)); % the lowest error
if e(m+1)<e(m) g = x( ind, 1:4); % the global best end
for n=1:I if x(n,5)<p(n,5) % the personal best p(n,:) = x(n,:); end v(n,:) = w*v(n,:) + c1*rand()*( p(n,1:4)-x(n,1:4)); v(n,:) = v(n,:) + c2*rand()*( g(1,1:4)-x(n,1:4)); x(n,1:4) = x(n,1:4) + dt*v(n,:); if x(n,1) > 9.00, x(n,1)=9.00; end % absorbing walls if x(n,2) > 0.05, x(n,2)=0.05; end if x(n,3) > 1.5, x(n,3)=1.5; end if x(n,4) > 2.2, x(n,4)=2.2; end end
end
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Global methodsparticle swarm optimization (5)
iter.
cost
0 10 20 30 400
1
2
3
4
5
6
7
[ 10 ]25
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Global methodsPSO (6)
cost [2]
A [mm]
B [mm]
h [mm]
eps [ – ]
Rin []
Xin []
534 5.481 0.050 1.46 1.57 176.9 -0.1
2 288 5.794 0.050 1.46 1.69 152.2 1.7
154 5.333 0.050 1.44 1.50 187.6 -0.5
21 5.406 0.050 1.48 1.54 196.7 3.2
50 iterations, 20 agents, c1 = c2 = 1.49, w = 0.9 -> 0.4, absorbing walls
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Searching for a minimumglobal first, local later
ITSS 2007, Pforzheim
f(x)
x
startingpoint
global local
Global optimization techniques …
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Searching for a minimumglobal first, local later
ITSS 2007, Pforzheim
f(x)
x
startingpoint
globallocal methodmethod
Global optimization techniques …
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Local minimizationgeneral algorithm (1)
1. Testing convergence. If the actual estimate of the optimum xk is accurate enough, then the algorithm is terminated. Otherwise, go to 2.
2. Computing search direction. Estimate the best direction pk moving the actual estimate of the optimum xk towards the optimum.
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Local minimizationgeneral algorithm (2)
3. Computing step length. Estimate scalar k ensuring the significant decrease of the value of the objective function: F(xk + kpk) < F(xk)
4. Updating the estimate of the minimum. Setxk+1 xk + k pk, k k + 1. Go back to 1.
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Testing algorithmsRosenbrock function
21
221221 1100, xxxxxF
function F = rosenbrock( x)
F = 100*( x(2,1) - x(1,1)^2)^2 +... ( 1 - x(1,1))^2;
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Steepest descentanalytical approach
kk gp kkkk gxx 1
function sda( alpha)
M = 10000;x = [ -1; +1];
for m=1:M g(1,1) = -400*x(1,1)*( x(2,1)-x(1,1)^2)-2*(1-x(1,1)); g(2,1) = 200*( x(2,1) - x(1,1)^2); x = x - alpha*g; out(m,:) = x';end
1k
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Steepest descentnumerical approach
function sdn( h)
M = 10000; alpha = 1e-3;x = [ -1; +1];
for m=1:M X1(1,1) = rosenbrock( [x(1,1) + h/2; x(2,1)]); X1(2,1) = rosenbrock( [x(1,1) - h/2; x(2,1)]); X2(1,1) = rosenbrock( [x(1,1); x(2,1) + h/2]); X2(2,1) = rosenbrock( [x(1,1); x(2,1) - h/2]); g(1,1) = (X1(1,1) - X1(2,1)) / h; g(2,1) = (X2(1,1) - X2(2,1)) / h; x = x - alpha*g; out(m,:) = x';end
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Newton methoddirection, step
pGppgpx kkkk FF T21T
kkk gGp 1
kkkk gGxx
11
x
y
x x x012
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Newton methodcode
function newton( x1, x2)
M = 10;x = [ x1; x2];
for m=1:M g(1,1) = -400*x(1,1)*(x(2,1)-x(1,1)^2)-2*(1-x(1,1)); g(2,1) = 200*( x(2,1) - x(1,1)^2); H(1,1) = 1200*x(1,1)^2 - 400*x(2,1) + 2; H(1,2) = -400*x(1,1); H(2,1) = -400*x(1,1); H(2,2) = 200; x = x - inv( H)*g; out(m,:) = x'end
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Steepest descent vs. Newtoncomparison
Steepest descent Newton method
• Properly chosen step length k
• Step length k = 1 all the time
• Convergence for Rosenbrock: 7000 steps
• Convergence for Rosenbrock: 3 steps
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ExampleGPS wire antenna
• Operation in frequency bands:– L1: central frequency fL1 = 1 575.4 MHz
– L2: central frequency fL2 = 1 227.6 MHz
• Omni-directional constant gain for the elevation from 5° to 90°
• Right-hand circular polarization
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GPS wire antennaGA v. PSO (1)
a)
iterations
F
b)
iterations
F
LUKEŠ, Z., RAIDA, Z. Multi-objective optimiza-tion of wire antennas: genetic algorithms versus particle swarm optimization. Radioengineering, 2005, vol. 14, no. 4, p. 91–97.
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a)
GPS wire antennaGA v. PSO (2) b)
LUKEŠ, Z., RAIDA, Z. Multi-objective optimiza-tion of wire antennas: genetic algorithms versus particle swarm optimization. Radioengineering, 2005, vol. 14, no. 4, p. 91–97.
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a)
GPS wire antennaGA v. PSO (3)
LUKEŠ, Z., RAIDA, Z. Multi-objective optimiza-tion of wire antennas: genetic algorithms versus particle swarm optimization. Radioengineering, 2005, vol. 14, no. 4, p. 91–97.
b)
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Z
a)
f[MHz]
GPS wire antennaGA v. PSO (4)
LUKEŠ, Z., RAIDA, Z. Multi-objective optimiza-tion of wire antennas: genetic algorithms versus particle swarm optimization. Radioengineering, 2005, vol. 14, no. 4, p. 91–97.
f[MHz]
Z
b)
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GPS wire antennaGA v. PSO (5)
a)
b)
a)
b)
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Conclusions
• Multi-objective optimization:a complex view on the structure
• Global optimization:perspective designs of a structure
• Local optimization:tuning of a relatively good design
ITSS 2007, Pforzheim Global optimization techniques …