Global Optimization Techniques in Computational Electromagnetics Zbyněk Raida Dept. of Radio Electronics Brno University of Technology Brno, Czechia

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

Optimization definition Searching for such values of state variables to meet desired parameters as close as possible ITSS 2007, PforzheimGlobal optimization techniques …

Optimization objective function (1) Deviation of the actual parameters of the system from the desired ones ITSS 2007, PforzheimGlobal optimization techniques …

Optimization objective function (2) ITSS 2007, PforzheimGlobal optimization techniques …

More objectives polarization 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. ITSS 2007, PforzheimGlobal optimization techniques …

More objectives polarization 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 Krakow (Poland), 2006, pp. 111–114. ITSS 2007, PforzheimGlobal optimization techniques …

More objectives directivity patterns (1) ITSS 2007, PforzheimGlobal optimization techniques …

More objectives directivity patterns (2)

More objectives multi-objective formulation ITSS 2007, PforzheimGlobal optimization techniques …

Multi-objective optimization two approaches ITSS 2007, PforzheimGlobal optimization techniques …

Searching for a minimum global versus local methods ITSS 2007, PforzheimGlobal optimization techniques …

Global methods genetic algorithms (1) ITSS 2007, PforzheimGlobal optimization techniques …

Global methods genetic algorithms (2) initial population quality evaluation selection ITSS 2007, Pforzheim

Global methods genetic algorithms (3) crossover mutation ITSS 2007, PforzheimGlobal optimization techniques …

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 resistance Xd = 0.0; % desired input reactance bit = [ ]; % bits per A, B, h, eps geb = norm( bit, 1) + 1; % bits in chromosome gen = round( rand( I, geb-1)); % 1st generation for 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);

Global methods genetic algorithms (5)

Global methods genetic algorithms (6) ITSS 2007, PforzheimGlobal optimization techniques …

Global methods particle 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. ITSS 2007, PforzheimGlobal optimization techniques …

Global methods PSO (2) ITSS 2007, Pforzheim

Global methods particle swarm optimization (3) absorbingreflectinginvisible ITSS 2007, PforzheimGlobal optimization techniques …

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 resistance Xd = 0; % required input reactance dt = 0.1; % time step c1 = 1.49; % personal scaling factor c2 = 1.49; % global scaling factor x = zeros( I, 5); % agents’ position p = zeros( I, 5); % personal best for n=1:I x(n,1) = *rand(); p(n,1) = x(n,1); x(n,2) = *rand(); p(n,2) = x(n,2); x(n,3) = * rand(); p(n,3) = x(n,3); x(n,4) = * rand(); p(n,4) = x(n,4); p(n,5) = 1e+6; end v = rand( I, 4); % agent velocity g = zeros( 1, 4); % global best e = zeros( G+1, 1); e(1) = 1e+6;

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

Global methods particle swarm optimization (5)

Global methods PSO (6) ITSS 2007, PforzheimGlobal optimization techniques …

Searching for a minimum global first, local later ITSS 2007, PforzheimGlobal optimization techniques …

Searching for a minimum global first, local later ITSS 2007, PforzheimGlobal optimization techniques …

Local minimization general algorithm (1) 1.Testing convergence. If the actual estimate of the optimum x k is accurate enough, then the algorithm is terminated. Otherwise, go to 2. 2.Computing search direction. Estimate the best direction p k moving the actual estimate of the optimum x k towards the optimum. ITSS 2007, PforzheimGlobal optimization techniques …

Local minimization general algorithm (2) 3.Computing step length. Estimate scalar  k ensuring the significant decrease of the value of the objective function: F(x k +  k p k ) < F(x k ) 4.Updating the estimate of the minimum. Set x k+1  x k +  k p k, k  k + 1. Go back to 1. ITSS 2007, PforzheimGlobal optimization techniques …

Testing algorithms Rosenbrock function function F = rosenbrock( x) F = 100*( x(2,1) - x(1,1)^2)^ ( 1 - x(1,1))^2; ITSS 2007, PforzheimGlobal optimization techniques …

Steepest descent analytical approach 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 ITSS 2007, PforzheimGlobal optimization techniques …

Steepest descent numerical 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 ITSS 2007, PforzheimGlobal optimization techniques …

Newton method direction, step ITSS 2007, PforzheimGlobal optimization techniques …

Newton method code 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)^ *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 ITSS 2007, PforzheimGlobal optimization techniques …

Steepest descent vs. Newton comparison Steepest descentNewton method Properly chosen step length  k Step length  k = 1 all the time Convergence for Rosenbrock:  7000 steps Convergence for Rosenbrock:  3 steps ITSS 2007, PforzheimGlobal optimization techniques …

Example GPS wire antenna Operation in frequency bands: –L1: central frequency f L1 = MHz –L2: central frequency f L2 = MHz Omni-directional constant gain for the elevation from 5° to 90° Right-hand circular polarization ITSS 2007, PforzheimGlobal optimization techniques …

GPS wire antenna GA v. PSO (1) 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. ITSS 2007, PforzheimGlobal optimization techniques …

GPS wire antenna GA v. PSO (2) 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. ITSS 2007, PforzheimGlobal optimization techniques …

GPS wire antenna GA 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. ITSS 2007, PforzheimGlobal optimization techniques …

GPS wire antenna GA 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. ITSS 2007, PforzheimGlobal optimization techniques …

GPS wire antenna GA v. PSO (5)

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, PforzheimGlobal optimization techniques …