Particle swarm optimization
Particle Swarm Optimization (PSO) is an optimization method in which multiple candidate solutions ('particles') migrate through the solution space under the influence of local and global best known positions. PSO does not require that the objective function be differentiable and can optimize over very large problem spaces, but is not guaranteed to converge. The method should be demonstrated by application of the functions recommended below, and possibly other standard or well-known optimization test cases.
The goal of parameter selection is to ensure that the global minimum is discriminated from any local minima, and that the minimum is accurately determined, and that convergence is achieved with acceptible resource usage. To provide a common basis for comparing implementations, the following test cases and parameter sets are recommended:
- McCormick function - bowl-shaped, with a single minimum; recommended parameters: omega = 0, phi p = 0.6, phi g = 0.3, number of particles = 100, number of iterations = 40
- Michalewicz function - steep ridges and valleys, with multiple minima; recommended parameters: omega = phi p = phi g = 0.3, number of particles = 1000, number of iterations = 30
References:
J
<lang J>load 'format/printf'
pso_init =: verb define
'Min Max parameters nParticles' =. y 'Min: %j\nMax: %j\nomega, phip, phig: %j\nnParticles: %j\n' printf Min;Max;parameters;nParticles nDims =. #Min pos =. Min +"1 (Max - Min) *"1 (nParticles,nDims) ?@$ 0 bpos =. pos bval =. (#pos) $ _ vel =. ($pos) $ 0 0;_;_;Min;Max;parameters;pos;vel;bpos;bval NB. initial state
)
pso =: adverb define
NB. previous state 'iter gbpos gbval Min Max parameters pos vel bpos0 bval' =. y
NB. evaluate val =. u"1 pos
NB. update better =. val < bval bpos =. (better # pos) (I. better)} bpos0 bval =. u"1 bpos gbval =. <./ bval gbpos =. bpos {~ (i. <./) bval
NB. migrate 'omega phip phig' =. parameters rp =. (#pos) ?@$ 0 rg =. ? 0 vel =. (omega*vel) + (phip * rp * bpos - pos) + (phig * rg * gbpos -"1 pos) pos =. pos + vel
NB. reset out-of-bounds particles bad =. +./"1 (Min >"1 pos) ,. (pos >"1 Max) newpos =. Min +"1 (Max-Min) *"1 ((+/bad),#Min) ?@$ 0 pos =. newpos (I. bad)} pos iter =. >: iter
NB. new state iter;gbpos;gbval;Min;Max;parameters;pos;vel;bpos;bval
)
reportState=: 'Iteration: %j\nGlobalBestPosition: %j\nGlobalBestValue: %j\n' printf 3&{.</lang> Apply to McCormick Function:<lang J> require 'trig'
mccormick =: sin@(+/) + *:@(-/) + 1 + _1.5 2.5 +/@:* ]
state =: pso_init _1.5 _3 ; 4 4 ; 0 0.6 0.3; 100
Min: _1.5 _3 Max: 4 4 omega, phip, phig: 0 0.6 0.3 nParticles: 100
state =: (mccormick pso)^:40 state reportState state
Iteration: 40 GlobalBestPosition: _0.547399 _1.54698 GlobalBestValue: _1.91322</lang> Apply to Michalewicz Function: <lang J> michalewicz =: 3 : '- +/ (sin y) * 20 ^~ sin (>: i. #y) * (*:y) % pi'
michalewicz =: [: -@(+/) sin * 20 ^~ sin@(pi %~ >:@i.@# * *:) NB. tacit equivalent state =: pso_init 0 0 ; (pi,pi) ; 0.3 0.3 0.3; 1000
Min: 0 0 Max: 3.14159 3.14159 omega, phip, phig: 0.3 0.3 0.3 nParticles: 1000
state =: (michalewicz pso)^:30 state reportState state
Iteration: 30 GlobalBestPosition: 2.20296 1.57083 GlobalBestValue: _1.8013</lang>
ooRexx
<lang oorexx>/* REXX ---------------------------------------------------------------
- Test for McCormick function
- --------------------------------------------------------------------*/
Numeric Digits 16 Parse Value '-.5 -1.5 1' With x y d fmin=1e9 Call refine x,y Do r=1 To 10
d=d/5 Call refine xmin,ymin End
Say 'which is better (less) than' Say ' f(-.54719,-1.54719)='f(-.54719,-1.54719) Say 'and differs from published -1.9133' Exit
refine: Parse Arg xx,yy Do x=xx-d To xx+d By d/2
Do y=yy-d To yy+d By d/2 f=f(x,y) If f<fmin Then Do Say x y f fmin=f xmin=x ymin=y End End End
Return
f: Parse Arg x,y res=rxcalcsin(x+y,16,'R')+(x-y)**2-1.5*x+2.5*y+1 Return res
- requires rxmath library</lang>
- Output:
-1.5 -2.5 -1.243197504692072 -1.0 -2.0 -1.641120008059867 -0.5 -1.5 -1.909297426825682 -0.54 -1.54 -1.913132979507516 -0.548 -1.548 -1.913221840016527 -0.5480 -1.5472 -1.913222034492829 -0.5472 -1.5472 -1.913222954970650 -0.54720000 -1.54719872 -1.913222954973731 -0.54719872 -1.54719872 -1.913222954978670 -0.54719872 -1.54719744 -1.913222954978914 -0.54719744 -1.54719744 -1.913222954981015 -0.5471975424 -1.5471975424 -1.913222954981036 which is better (less) than f(-.54719,-1.54719)=-1.913222954882273 and differs from published -1.9133
REXX
This REXX version uses a large numeric digits (but only displays 16 digits).
The numeric precision is only limited to the number of decimal digits in the pi variable (in this case, 77). <lang rexx>/*REXX pgm calc. Particle Swarm Optimization as it migrates through a solution*/ numeric digits length(pi()); sDig=16 /*SDIG: the number of displayed digits.*/ parse arg x y d p . /*obtain optional arguments from the CL*/ if x== | x==',' then x= -0.5 /*X not defined? Then use the default.*/ if y== | y==',' then y= -1.5 /*Y " " " " " " */ if d== | d==',' then d= 1 /*D " " " " " " */ if p== | p==',' then p= 1e12 /*P " " " " " " */ minF=p /*P the number of particles: 1 billion*/ say center('X', sDig+3, '═') center('Y', sDig+3, '═') center('D', sDig+3, '═') call refine x,y
do r=1 for 10; d=d*.5 call refine minX, minY end /*r*/
say say 'Which is better (less) than the global minimum at:' say ' f(-.54719, -1.54719) ───► ' fmt(f(-.54719, -1.54719)) say 'The published global minimum is: -1.9133' exit /*────────────────────────────────────────────────────────────────────────────*/ refine: parse arg xx,yy; dh=d * 0.5
do x=xx-d to xx+d by dh do y=yy-d to yy+d by dh; f=f(x,y); if f>=minF then iterate say fmt(x) fmt(y) fmt(f); minF=f; minX=x; minY=y end /*y*/ end /*x*/
return /*────────────────────────────────────────────────────────────────────────────*/ fmt: parse arg ?; ?=format(?,,sDig) /*format number with Sdig decimal digs.*/ L=length(?); if pos(.,?)\==0 then ?=strip(strip(?,'T',0),'T',.);return left(?,L) /*────────────────────────────────────────────────────────────────────────────*/ sin: procedure; parse arg x; x=r2r(x); numeric fuzz 5; z=x; _=x; q=x*x
do k=2 by 2 until p=z; p=z; _=-_*q/(k*(k+1)); z=z+_; end; return z
/*──────────────────────────────────one─liner subroutines──────────────────────────────────────*/ f: procedure: parse arg a,b; return sin(a+b) + (a-b)**2 - 1.5*a + 2.5*b + 1 pi: pi=3.1415926535897932384626433832795028841971693993751058209749445923078164062862; return pi r2r: return arg(1) // (pi()*2) /*normalize radians ───► a unit circle.*/</lang> output when using the default inputs:
═════════X═════════ ═════════Y═════════ ═════════D═════════ -1.5 -2.5 -1.2431975046920717 -1 -2 -1.6411200080598672 -0.5 -1.5 -1.9092974268256817 -0.5625 -1.5625 -1.912819789818452 -0.5625 -1.546875 -1.9128819293954732 -0.546875 -1.546875 -1.9132227747573614 -0.54736328125 -1.54736328125 -1.9132229074107836 Which is better (less) than the global minimum at: f(-.54719, -1.54719) ───► -1.9132229548822736 The published global minimum is: -1.9133