Particle swarm optimization: Difference between revisions

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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 are recommended:

McCormick function - bowl-shaped, with a single minimum
- -1.5 < x1 < 4
- -3 < x2 < 4
- 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
- 0 < x1 < pi
- 0 < x2 < pi
- omega = 0.3
- phi p = 0.3
- phi g = 0.3
- number of particles = 1000
- number of iterations = 30

References:

[Particle Swarm Optimization[1]]
[Virtual Library of Optimization Test Functions[2]]

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

Translation of: ooRexx

This REXX version uses a large numeric digits (but only displays 25 digits).

Classic REXX doesn't have a sine function, so a RYO version is included here.

The numeric precision is only limited to the number of decimal digits defined in the pi variable (in this case, 100).

This REXX version supports the specifying of X, Y, and D, as well as the number of particles, the number of times the
computation loop is performed, and the number of decimal digits to be displayed.

The refinement loop is stopped when the function value stabilizes, or the limit of iterations is reached. <lang rexx>/*REXX pgm calc. Particle Swarm Optimization as it migrates through a solution*/ numeric digits length(pi()) /*sDigs: is the # of displayed digits.*/ parse arg x y d #part times sDigs . /*obtain optional arguments from the CL*/ if x== | x==',' then x= -0.5 /*is X not defined?*/ if y== | y==',' then y= -1.5 /* " Y " " */ if d== | d==',' then d= 1 /* " D " " */ if #part== | #part==',' then #part=1e12 /* " the # particles " " */ if times== | times==',' then times= 40 /* " the # of times " " */ if sDigs== | sDigs==',' then sDigs= 25 /* " the # of digits " " */ minF=#part /*number of particles is one billion. */ say center('X',sDigs+3,'═') center('Y',sDigs+3,'═') center('D',sDigs+3,'═') call refine x,y

               do stuff=1  for times  until old=f;      d=d*.2;       old=f
               call refine minX, minY
               end   /*stuff*/        /* [↑]  stop refining after TIMES,  or */

say /* when the value of F stabilizes.*/ indent=1 + 2*(sDigs+3) /*compute the indentation for alignment*/ say right('The global minimum for f(-.54719, -1.54719) ───► ', indent) fmt(f(-.54719, -1.54719)) say right('The published global minimum is:' , indent) fmt( -1.9133 ) exit /*stick a fork in it, we're all done. */ /*────────────────────────────────────────────────────────────────────────────*/ 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 /*──────────────────────────────────────────────────────────────────────────────────one─liner subroutines───────────────────────────────*/ f: procedure: parse arg a,b; return sin(a+b) + (a-b)**2 - 1.5*a + 2.5*b + 1 fmt: parse arg ?; ?=format(?,,sDigs); L=length(?); if pos(.,?)\==0 then ?=strip(strip(?,'T',0),'T',.); return left(?,L) pi: pi=3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068; return pi r2r: return arg(1) // (pi()*2) /*normalize radians ───► a unit circle.*/ 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</lang> output when using the default inputs:

═════════════X══════════════ ═════════════Y══════════════ ═════════════D══════════════
-1.5                         -2.5                         -1.2431975046920717486273609
-1                           -2                           -1.6411200080598672221007448
-0.5                         -1.5                         -1.9092974268256816953960199
-0.54                        -1.54                        -1.9131329795075164948766768
-0.548                       -1.548                       -1.9132218400165267634506035
-0.548                       -1.5472                      -1.9132220344928294065568196
-0.5472                      -1.5472                      -1.9132229549706499208388746
-0.5472                      -1.54719872                  -1.9132229549737311254290577
-0.54719872                  -1.54719872                  -1.9132229549786702369612333
-0.54719872                  -1.54719744                  -1.91322295497891365438682
-0.54719744                  -1.54719744                  -1.9132229549810149766572388
-0.5471975424                -1.5471975424                -1.9132229549810362588916172
-0.54719755264               -1.54719755264               -1.9132229549810363893093655
-0.547197550592              -1.547197550592              -1.9132229549810363922848065
-0.5471975514112             -1.5471975514112             -1.9132229549810363928381695
-0.5471975510016             -1.5471975510016             -1.9132229549810363928520779
-0.54719755116544            -1.54719755116544            -1.9132229549810363929162561
-0.547197551198208           -1.547197551198208           -1.9132229549810363929179331
-0.547197551198208           -1.54719755119755264         -1.9132229549810363929179344
-0.54719755119755264         -1.54719755119755264         -1.9132229549810363929179361
-0.54719755119755264         -1.54719755119689728         -1.9132229549810363929179365
-0.54719755119689728         -1.54719755119689728         -1.9132229549810363929179375
-0.54719755119689728         -1.547197551196766208        -1.9132229549810363929179375
-0.547197551196766208        -1.547197551196766208        -1.9132229549810363929179376
-0.547197551196766208        -1.547197551196635136        -1.9132229549810363929179376
-0.547197551196635136        -1.547197551196635136        -1.9132229549810363929179376
-0.547197551196635136        -1.5471975511966089216       -1.9132229549810363929179376
-0.5471975511966089216       -1.5471975511966089216       -1.9132229549810363929179376
-0.5471975511966089216       -1.54719755119660367872      -1.9132229549810363929179376
-0.54719755119660367872      -1.54719755119660367872      -1.9132229549810363929179376
-0.54719755119660367872      -1.54719755119659843584      -1.9132229549810363929179376
-0.54719755119659843584      -1.54719755119659843584      -1.9132229549810363929179376
-0.547197551196597387264     -1.547197551196597387264     -1.9132229549810363929179376
-0.5471975511965978066944    -1.5471975511965978066944    -1.9132229549810363929179376
-0.5471975511965978066944    -1.54719755119659776475136   -1.9132229549810363929179376
-0.54719755119659776475136   -1.54719755119659776475136   -1.9132229549810363929179376
-0.54719755119659776475136   -1.547197551196597756362752  -1.9132229549810363929179376
-0.547197551196597756362752  -1.547197551196597756362752  -1.9132229549810363929179376
-0.547197551196597756362752  -1.547197551196597747974144  -1.9132229549810363929179376
-0.547197551196597747974144  -1.547197551196597747974144  -1.9132229549810363929179376
-0.547197551196597747974144  -1.5471975511965977462964224 -1.9132229549810363929179376
-0.5471975511965977462964224 -1.5471975511965977462964224 -1.9132229549810363929179376
-0.5471975511965977462964224 -1.5471975511965977462293135 -1.9132229549810363929179376
-0.5471975511965977462293135 -1.5471975511965977462293135 -1.9132229549810363929179376
-0.5471975511965977462293135 -1.5471975511965977461622047 -1.9132229549810363929179376
-0.5471975511965977461622047 -1.5471975511965977461622047 -1.9132229549810363929179376
-0.5471975511965977461487829 -1.5471975511965977461487829 -1.9132229549810363929179376
-0.5471975511965977461541516 -1.5471975511965977461541516 -1.9132229549810363929179376
-0.547197551196597746154259  -1.547197551196597746154259  -1.9132229549810363929179376
-0.547197551196597746154259  -1.5471975511965977461542375 -1.9132229549810363929179376
-0.5471975511965977461542375 -1.5471975511965977461542375 -1.9132229549810363929179376
-0.5471975511965977461542375 -1.547197551196597746154216  -1.9132229549810363929179376
-0.547197551196597746154216  -1.547197551196597746154216  -1.9132229549810363929179376
-0.547197551196597746154216  -1.5471975511965977461542152 -1.9132229549810363929179376
-0.5471975511965977461542152 -1.5471975511965977461542152 -1.9132229549810363929179376
-0.5471975511965977461542152 -1.5471975511965977461542143 -1.9132229549810363929179376
-0.5471975511965977461542143 -1.5471975511965977461542143 -1.9132229549810363929179376
-0.5471975511965977461542145 -1.5471975511965977461542145 -1.9132229549810363929179376
-0.5471975511965977461542145 -1.5471975511965977461542145 -1.9132229549810363929179376
-0.5471975511965977461542145 -1.5471975511965977461542145 -1.9132229549810363929179376
-0.5471975511965977461542145 -1.5471975511965977461542145 -1.9132229549810363929179376
-0.5471975511965977461542145 -1.5471975511965977461542145 -1.9132229549810363929179376
-0.5471975511965977461542145 -1.5471975511965977461542145 -1.9132229549810363929179376
-0.5471975511965977461542145 -1.5471975511965977461542145 -1.9132229549810363929179376

      The global minimum for  f(-.54719, -1.54719)  ───►  -1.9132229548822735814541188
                         The published global minimum is: -1.9133

Output note: the published global minimum (referenced above, as well as the function's arguments) can be found at:

http://www.sfu.ca/~ssurjano/mccorm.html