Particle Swarm Optimization

From Rosetta Code
Particle Swarm Optimization is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

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
      function parameters and bounds (recommended):
    • -1.5 < x1 < 4
    • -3 < x2 < 4
      search parameters (suggested):
    • 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
      function parameters and bounds (recommended):
    • 0 < x1 < pi
    • 0 < x2 < pi
      search parameters (suggested):
    • 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]]



C#[edit]

Translation of: java
using System;
 
namespace ParticleSwarmOptimization {
public struct Parameters {
public double omega, phip, phig;
 
public Parameters(double omega, double phip, double phig) : this() {
this.omega = omega;
this.phip = phip;
this.phig = phig;
}
}
 
public struct State {
public int iter;
public double[] gbpos;
public double gbval;
public double[] min;
public double[] max;
public Parameters parameters;
public double[][] pos;
public double[][] vel;
public double[][] bpos;
public double[] bval;
public int nParticles;
public int nDims;
 
public State(int iter, double[] gbpos, double gbval, double[] min, double[] max, Parameters parameters, double[][] pos, double[][] vel, double[][] bpos, double[] bval, int nParticles, int nDims) : this() {
this.iter = iter;
this.gbpos = gbpos;
this.gbval = gbval;
this.min = min;
this.max = max;
this.parameters = parameters;
this.pos = pos;
this.vel = vel;
this.bpos = bpos;
this.bval = bval;
this.nParticles = nParticles;
this.nDims = nDims;
}
 
public void Report(string testfunc) {
Console.WriteLine("Test Function  : {0}", testfunc);
Console.WriteLine("Iterations  : {0}", iter);
Console.WriteLine("Global Best Position : {0}", string.Join(", ", gbpos));
Console.WriteLine("Global Best Value  : {0}", gbval);
}
}
 
class Program {
public static State PsoInit(double[] min, double[] max, Parameters parameters, int nParticles) {
var nDims = min.Length;
double[][] pos = new double[nParticles][];
for (int i = 0; i < nParticles; i++) {
pos[i] = new double[min.Length];
min.CopyTo(pos[i], 0);
}
double[][] vel = new double[nParticles][];
for (int i = 0; i < nParticles; i++) {
vel[i] = new double[nDims];
}
double[][] bpos = new double[nParticles][];
for (int i = 0; i < nParticles; i++) {
bpos[i] = new double[min.Length];
min.CopyTo(bpos[i], 0);
}
double[] bval = new double[nParticles];
for (int i = 0; i < nParticles; i++) {
bval[i] = double.PositiveInfinity;
}
int iter = 0;
double[] gbpos = new double[nDims];
for (int i = 0; i < nDims; i++) {
gbpos[i] = double.PositiveInfinity;
}
double gbval = double.PositiveInfinity;
 
return new State(iter, gbpos, gbval, min, max, parameters, pos, vel, bpos, bval, nParticles, nDims);
}
 
static Random r = new Random();
 
public static State Pso(Func<double[], double> fn, State y) {
var p = y.parameters;
double[] v = new double[y.nParticles];
double[][] bpos = new double[y.nParticles][];
for (int i = 0; i < y.nParticles; i++) {
bpos[i] = new double[y.min.Length];
y.min.CopyTo(bpos[i], 0);
}
double[] bval = new double[y.nParticles];
double[] gbpos = new double[y.nDims];
double gbval = double.PositiveInfinity;
for (int j = 0; j < y.nParticles; j++) {
// evaluate
v[j] = fn.Invoke(y.pos[j]);
// update
if (v[j] < y.bval[j]) {
y.pos[j].CopyTo(bpos[j], 0);
bval[j] = v[j];
}
else {
y.bpos[j].CopyTo(bpos[j], 0);
bval[j] = y.bval[j];
}
if (bval[j] < gbval) {
gbval = bval[j];
bpos[j].CopyTo(gbpos, 0);
}
}
double rg = r.NextDouble();
double[][] pos = new double[y.nParticles][];
double[][] vel = new double[y.nParticles][];
for (int i = 0; i < y.nParticles; i++) {
pos[i] = new double[y.nDims];
vel[i] = new double[y.nDims];
}
for (int j = 0; j < y.nParticles; j++) {
// migrate
double rp = r.NextDouble();
bool ok = true;
for (int k = 0; k < y.nDims; k++) {
vel[j][k] = 0.0;
pos[j][k] = 0.0;
}
for (int k = 0; k < y.nDims; k++) {
vel[j][k] = p.omega * y.vel[j][k] +
p.phip * rp * (bpos[j][k] - y.pos[j][k]) +
p.phig * rg * (gbpos[k] - y.pos[j][k]);
pos[j][k] = y.pos[j][k] + vel[j][k];
ok = ok && y.min[k] < pos[j][k] && y.max[k] > pos[j][k];
}
if (!ok) {
for (int k = 0; k < y.nDims; k++) {
pos[j][k] = y.min[k] + (y.max[k] - y.min[k]) * r.NextDouble();
}
}
}
var iter = 1 + y.iter;
return new State(iter, gbpos, gbval, y.min, y.max, y.parameters, pos, vel, bpos, bval, y.nParticles, y.nDims);
}
 
public static State Iterate(Func<double[], double> fn, int n, State y) {
State r = y;
if (n == int.MaxValue) {
State old = y;
while (true) {
r = Pso(fn, r);
if (r.Equals(old)) break;
old = r;
}
}
else {
for (int i = 0; i < n; i++) {
r = Pso(fn, r);
}
}
return r;
}
 
public static double Mccormick(double[] x) {
var a = x[0];
var b = x[1];
return Math.Sin(a + b) + (a - b) * (a - b) + 1.0 + 2.5 * b - 1.5 * a;
}
 
public static double Michalewicz(double[] x) {
int m = 10;
int d = x.Length;
double sum = 0.0;
for (int i = 1; i < d; i++) {
var j = x[i - 1];
var k = Math.Sin(i * j * j / Math.PI);
sum += Math.Sin(j) * Math.Pow(k, 2.0 * m);
}
return -sum;
}
 
static void Main(string[] args) {
var state = PsoInit(
new double[] { -1.5, -3.0 },
new double[] { 4.0, 4.0 },
new Parameters(0.0, 0.6, 0.3),
100
);
state = Iterate(Mccormick, 40, state);
state.Report("McCormick");
Console.WriteLine("f(-.54719, -1.54719) : {0}", Mccormick(new double[] { -.54719, -1.54719 }));
Console.WriteLine();
 
state = PsoInit(
new double[] { -0.0, -0.0 },
new double[] { Math.PI, Math.PI },
new Parameters(0.3, 0.3, 0.3),
1000
);
state = Iterate(Michalewicz, 30, state);
state.Report("Michalewicz (2D)");
Console.WriteLine("f(2.20, 1.57)  : {0}", Michalewicz(new double[] { 2.20, 1.57 }));
}
}
}
Output:
Test Function        : McCormick
Iterations           : 40
Global Best Position : -0.546850526417689, -1.54649614884518
Global Best Value    : -1.91322235333426
f(-.54719, -1.54719) : -1.91322295488227

Test Function        : Michalewicz (2D)
Iterations           : 30
Global Best Position : 2.20290514143486, 2.20798457238775
Global Best Value    : -0.801303410096221
f(2.20, 1.57)        : -0.801166387820286

D[edit]

Translation of: Kotlin
import std.math;
import std.random;
import std.stdio;
 
alias Func = double function(double[]);
 
struct Parameters {
double omega, phip, phig;
}
 
struct State {
int iter;
double[] gbpos;
double gbval;
double[] min;
double[] max;
Parameters parameters;
double[][] pos;
double[][] vel;
double[][] bpos;
double[] bval;
int nParticles;
int nDims;
 
void report(string testfunc) {
writeln("Test Function  : ", testfunc);
writeln("Iterations  : ", iter);
writefln("Global Best Position : [%(%.16f, %)]", gbpos);
writefln("Global Best Value  : %.16f", gbval);
}
}
 
State psoInit(double[] min, double[] max, Parameters parameters, int nParticles) {
auto nDims = min.length;
double[][] pos;
pos.length = nParticles;
pos[] = min;
double[][] vel;
vel.length = nParticles;
for (int i; i<nParticles; i++) vel[i].length = nDims;
double[][] bpos;
bpos.length = nParticles;
bpos[] = min;
double[] bval;
bval.length = nParticles;
bval[] = double.infinity;
auto iter = 0;
double[] gbpos;
gbpos.length = nDims;
gbpos[] = double.infinity;
auto gbval = double.infinity;
return State(iter, gbpos, gbval, min, max, parameters, pos, vel, bpos, bval, nParticles, nDims);
}
 
State pso(Func fn, State y) {
auto p = y.parameters;
double[] v;
v.length = y.nParticles;
double[][] bpos;
bpos.length = y.nParticles;
bpos[] = y.min;
double[] bval;
bval.length = y.nParticles;
double[] gbpos;
gbpos.length = y.nDims;
auto gbval = double.infinity;
foreach (j; 0..y.nParticles) {
// evaluate
v[j] = fn(y.pos[j]);
// update
if (v[j] < y.bval[j]) {
bpos[j] = y.pos[j];
bval[j] = v[j];
} else {
bpos[j] = y.bpos[j];
bval[j] = y.bval[j];
}
if (bval[j] < gbval) {
gbval = bval[j];
gbpos = bpos[j];
}
}
auto rg = uniform01();
double[][] pos;
pos.length = y.nParticles;
for (int i; i<pos.length; i++) pos[i].length = y.nDims;
double[][] vel;
vel.length = y.nParticles;
for (int i; i<vel.length; i++) vel[i].length = y.nDims;
foreach (j; 0..y.nParticles) {
// migrate
auto rp = uniform01();
bool ok = true;
vel[j][] = 0;
pos[j][] = 0;
foreach (k; 0..y.nDims) {
vel[j][k] = p.omega * y.vel[j][k] +
p.phip * rp * (bpos[j][k] - y.pos[j][k]) +
p.phig * rg * (gbpos[k] - y.pos[j][k]);
pos[j][k] = y.pos[j][k] + vel[j][k];
ok = ok && y.min[k] < pos[j][k] && y.max[k] > pos[j][k];
}
if (!ok) {
foreach (k; 0..y.nDims) {
pos[j][k] = y.min[k] + (y.max[k] - y.min[k]) * uniform01();
}
}
}
auto iter = 1 + y.iter;
return State(iter, gbpos, gbval, y.min, y.max, y.parameters, pos, vel, bpos, bval, y.nParticles, y.nDims);
}
 
State iterate(Func fn, int n, State y) {
auto r = y;
auto old = y;
if (n == int.max) {
while (true) {
r = pso(fn, r);
if (r == old) break;
old = r;
}
} else {
foreach (_; 0..n) r = pso(fn, r);
}
return r;
}
 
double mccormick(double[] x) {
auto a = x[0];
auto b = x[1];
return sin(a + b) + (a - b) * (a - b) + 1.0 + 2.5 * b - 1.5 * a;
}
 
double michalewicz(double[] x) {
auto m = 10;
auto d = x.length;
auto sum = 0.0;
foreach (i; 1..d) {
auto j = x[i - 1];
auto k = sin(i * j * j / PI);
sum += sin(j) * k^^(2.0*m);
}
return -sum;
}
 
void main() {
auto state = psoInit(
[-1.5, -3.0],
[4.0, 4.0],
Parameters(0.0, 0.6, 0.3),
100
);
state = iterate(&mccormick, 40, state);
state.report("McCormick");
writefln("f(-.54719, -1.54719) : %.16f", mccormick([-.54719, -1.54719]));
writeln;
state = psoInit(
[0.0, 0.0],
[PI, PI],
Parameters(0.3, 0.3, 0.3),
1000
);
state = iterate(&michalewicz, 30, state);
state.report("Michalewicz (2D)");
writefln("f(2.20, 1.57)  : %.16f", michalewicz([2.2, 1.57]));
}
Output:
Test Function        : McCormick
Iterations           : 40
Global Best Position : [-0.5673174452967942, -1.5373177402652800]
Global Best Value    : -1.9122776571457756
f(-.54719, -1.54719) : -1.9132229548822735

Test Function        : Michalewicz (2D)
Iterations           : 30
Global Best Position : [2.1907380816516597, 1.5608474620076016]
Global Best Value    : -1.7949374368688056
f(2.20, 1.57)        : -1.8011407184738251

Go[edit]

Translation of: Kotlin
package main
 
import (
"fmt"
"math"
"math/rand"
"time"
)
 
type ff = func([]float64) float64
 
type parameters struct{ omega, phip, phig float64 }
 
type state struct {
iter int
gbpos []float64
gbval float64
min []float64
max []float64
params parameters
pos [][]float64
vel [][]float64
bpos [][]float64
bval []float64
nParticles int
nDims int
}
 
func (s state) report(testfunc string) {
fmt.Println("Test Function  :", testfunc)
fmt.Println("Iterations  :", s.iter)
fmt.Println("Global Best Position :", s.gbpos)
fmt.Println("Global Best Value  :", s.gbval)
}
 
func psoInit(min, max []float64, params parameters, nParticles int) *state {
nDims := len(min)
pos := make([][]float64, nParticles)
vel := make([][]float64, nParticles)
bpos := make([][]float64, nParticles)
bval := make([]float64, nParticles)
for i := 0; i < nParticles; i++ {
pos[i] = min
vel[i] = make([]float64, nDims)
bpos[i] = min
bval[i] = math.Inf(1)
}
iter := 0
gbpos := make([]float64, nDims)
for i := 0; i < nDims; i++ {
gbpos[i] = math.Inf(1)
}
gbval := math.Inf(1)
return &state{iter, gbpos, gbval, min, max, params,
pos, vel, bpos, bval, nParticles, nDims}
}
 
func pso(fn ff, y *state) *state {
p := y.params
v := make([]float64, y.nParticles)
bpos := make([][]float64, y.nParticles)
bval := make([]float64, y.nParticles)
gbpos := make([]float64, y.nDims)
gbval := math.Inf(1)
for j := 0; j < y.nParticles; j++ {
// evaluate
v[j] = fn(y.pos[j])
// update
if v[j] < y.bval[j] {
bpos[j] = y.pos[j]
bval[j] = v[j]
} else {
bpos[j] = y.bpos[j]
bval[j] = y.bval[j]
}
if bval[j] < gbval {
gbval = bval[j]
gbpos = bpos[j]
}
}
rg := rand.Float64()
pos := make([][]float64, y.nParticles)
vel := make([][]float64, y.nParticles)
for j := 0; j < y.nParticles; j++ {
pos[j] = make([]float64, y.nDims)
vel[j] = make([]float64, y.nDims)
// migrate
rp := rand.Float64()
ok := true
for z := 0; z < y.nDims; z++ {
pos[j][z] = 0
vel[j][z] = 0
}
for k := 0; k < y.nDims; k++ {
vel[j][k] = p.omega*y.vel[j][k] +
p.phip*rp*(bpos[j][k]-y.pos[j][k]) +
p.phig*rg*(gbpos[k]-y.pos[j][k])
pos[j][k] = y.pos[j][k] + vel[j][k]
ok = ok && y.min[k] < pos[j][k] && y.max[k] > pos[j][k]
}
if !ok {
for k := 0; k < y.nDims; k++ {
pos[j][k] = y.min[k] + (y.max[k]-y.min[k])*rand.Float64()
}
}
}
iter := 1 + y.iter
return &state{iter, gbpos, gbval, y.min, y.max, y.params,
pos, vel, bpos, bval, y.nParticles, y.nDims}
}
 
func iterate(fn ff, n int, y *state) *state {
r := y
for i := 0; i < n; i++ {
r = pso(fn, r)
}
return r
}
 
func mccormick(x []float64) float64 {
a, b := x[0], x[1]
return math.Sin(a+b) + (a-b)*(a-b) + 1.0 + 2.5*b - 1.5*a
}
 
func michalewicz(x []float64) float64 {
m := 10.0
sum := 0.0
for i := 1; i <= len(x); i++ {
j := x[i-1]
k := math.Sin(float64(i) * j * j / math.Pi)
sum += math.Sin(j) * math.Pow(k, 2*m)
}
return -sum
}
 
func main() {
rand.Seed(time.Now().UnixNano())
st := psoInit(
[]float64{-1.5, -3.0},
[]float64{4.0, 4.0},
parameters{0.0, 0.6, 0.3},
100,
)
st = iterate(mccormick, 40, st)
st.report("McCormick")
fmt.Println("f(-.54719, -1.54719) :", mccormick([]float64{-.54719, -1.54719}))
fmt.Println()
st = psoInit(
[]float64{0.0, 0.0},
[]float64{math.Pi, math.Pi},
parameters{0.3, 0.3, 0.3},
1000,
)
st = iterate(michalewicz, 30, st)
st.report("Michalewicz (2D)")
fmt.Println("f(2.20, 1.57)  :", michalewicz([]float64{2.2, 1.57}))
 
Output:

Sample output:

Test Function        : McCormick
Iterations           : 40
Global Best Position : [-0.5473437041724806 -1.5464923165739348]
Global Best Value    : -1.9132220947578635
f(-.54719, -1.54719) : -1.913222954882274

Test Function        : Michalewicz (2D)
Iterations           : 30
Global Best Position : [2.2029051150895165 1.570796212894911]
Global Best Value    : -1.8013034100953598
f(2.20, 1.57)        : -1.8011407184738253

J[edit]

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) [email protected]$ 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) [email protected]$ 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) [email protected]$ 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&{.
Apply to McCormick Function:
   require 'trig'
mccormick =: [email protected](+/) + *:@(-/) + 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

Apply to Michalewicz Function:

   michalewicz =: 3 : '- +/ (sin y) * 20 ^~ sin (>: i. #y) * (*:y) % pi'
michalewicz =: [: [email protected](+/) sin * 20 ^~ [email protected](pi %~ >:@[email protected]# * *:) 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

JavaScript[edit]

Translation of J.

function pso_init(y) {
var nDims= y.min.length;
var pos=[], vel=[], bpos=[], bval=[];
for (var j= 0; j<y.nParticles; j++) {
pos[j]= bpos[j]= y.min;
var v= []; for (var k= 0; k<nDims; k++) v[k]= 0;
vel[j]= v;
bval[j]= Infinity}
return {
iter: 0,
gbpos: Infinity,
gbval: Infinity,
min: y.min,
max: y.max,
parameters: y.parameters,
pos: pos,
vel: vel,
bpos: bpos,
bval: bval,
nParticles: y.nParticles,
nDims: nDims}
}
 
function pso(fn, state) {
var y= state;
var p= y.parameters;
var val=[], bpos=[], bval=[], gbval= Infinity, gbpos=[];
for (var j= 0; j<y.nParticles; j++) {
// evaluate
val[j]= fn.apply(null, y.pos[j]);
// update
if (val[j] < y.bval[j]) {
bpos[j]= y.pos[j];
bval[j]= val[j];
} else {
bpos[j]= y.bpos[j];
bval[j]= y.bval[j]}
if (bval[j] < gbval) {
gbval= bval[j];
gbpos= bpos[j]}}
var rg= Math.random(), vel=[], pos=[];
for (var j= 0; j<y.nParticles; j++) {
// migrate
var rp= Math.random(), ok= true;
vel[j]= [];
pos[j]= [];
for (var k= 0; k < y.nDims; k++) {
vel[j][k]= p.omega*y.vel[j][k] + p.phip*rp*(bpos[j]-y.pos[j]) + p.phig*rg*(gbpos-y.pos[j]);
pos[j][k]= y.pos[j]+vel[j][k];
ok= ok && y.min[k]<pos[j][k] && y.max>pos[j][k];}
if (!ok)
for (var k= 0; k < y.nDims; k++)
pos[j][k]= y.min[k] + (y.max[k]-y.min[k])*Math.random()}
return {
iter: 1+y.iter,
gbpos: gbpos,
gbval: gbval,
min: y.min,
max: y.max,
parameters: y.parameters,
pos: pos,
vel: vel,
bpos: bpos,
bval: bval,
nParticles: y.nParticles,
nDims: y.nDims}
}
 
function display(text) {
if (document) {
var o= document.getElementById('o');
if (!o) {
o= document.createElement('pre');
o.id= 'o';
document.body.appendChild(o)}
o.innerHTML+= text+'\n';
window.scrollTo(0,document.body.scrollHeight);
}
if (console.log) console.log(text)
}
 
function reportState(state) {
var y= state;
display('');
display('Iteration: '+y.iter);
display('GlobalBestPosition: '+y.gbpos);
display('GlobalBestValue: '+y.gbval);
}
 
function repeat(fn, n, y) {
var r=y, old= y;
if (Infinity == n)
while ((r= fn(r)) != old) old= r;
else
for (var j= 0; j<n; j++) r= fn(r);
return r
}
 
function mccormick(a,b) {
return Math.sin(a+b) + Math.pow(a-b,2) + (1 + 2.5*b - 1.5*a)
}
 
state= pso_init({
min: [-1.5,-3], max:[4,4],
parameters: {omega: 0, phip: 0.6, phig: 0.3},
nParticles: 100});
 
reportState(state);
 
state= repeat(function(y){return pso(mccormick,y)}, 40, state);
 
reportState(state);

Example displayed result (random numbers are involved so there will be a bit of variance between repeated runs:

 
Iteration: 0
GlobalBestPosition: Infinity
GlobalBestValue: Infinity
 
Iteration: 40
GlobalBestPosition: -0.5134004259016365,-1.5512442672625184
GlobalBestValue: -1.9114053788600853

Java[edit]

Translation of: Kotlin
import java.util.Arrays;
import java.util.Objects;
import java.util.Random;
import java.util.function.Function;
 
public class App {
static class Parameters {
double omega;
double phip;
double phig;
 
Parameters(double omega, double phip, double phig) {
this.omega = omega;
this.phip = phip;
this.phig = phig;
}
}
 
static class State {
int iter;
double[] gbpos;
double gbval;
double[] min;
double[] max;
Parameters parameters;
double[][] pos;
double[][] vel;
double[][] bpos;
double[] bval;
int nParticles;
int nDims;
 
State(int iter, double[] gbpos, double gbval, double[] min, double[] max, Parameters parameters, double[][] pos, double[][] vel, double[][] bpos, double[] bval, int nParticles, int nDims) {
this.iter = iter;
this.gbpos = gbpos;
this.gbval = gbval;
this.min = min;
this.max = max;
this.parameters = parameters;
this.pos = pos;
this.vel = vel;
this.bpos = bpos;
this.bval = bval;
this.nParticles = nParticles;
this.nDims = nDims;
}
 
void report(String testfunc) {
System.out.printf("Test Function  : %s\n", testfunc);
System.out.printf("Iterations  : %d\n", iter);
System.out.printf("Global Best Position : %s\n", Arrays.toString(gbpos));
System.out.printf("Global Best value  : %.15f\n", gbval);
}
}
 
private static State psoInit(double[] min, double[] max, Parameters parameters, int nParticles) {
int nDims = min.length;
double[][] pos = new double[nParticles][];
for (int i = 0; i < nParticles; ++i) {
pos[i] = min.clone();
}
double[][] vel = new double[nParticles][nDims];
double[][] bpos = new double[nParticles][];
for (int i = 0; i < nParticles; ++i) {
bpos[i] = min.clone();
}
double[] bval = new double[nParticles];
for (int i = 0; i < bval.length; ++i) {
bval[i] = Double.POSITIVE_INFINITY;
}
int iter = 0;
double[] gbpos = new double[nDims];
for (int i = 0; i < gbpos.length; ++i) {
gbpos[i] = Double.POSITIVE_INFINITY;
}
double gbval = Double.POSITIVE_INFINITY;
return new State(iter, gbpos, gbval, min, max, parameters, pos, vel, bpos, bval, nParticles, nDims);
}
 
private static Random r = new Random();
 
private static State pso(Function<double[], Double> fn, State y) {
Parameters p = y.parameters;
double[] v = new double[y.nParticles];
double[][] bpos = new double[y.nParticles][];
for (int i = 0; i < y.nParticles; ++i) {
bpos[i] = y.min.clone();
}
double[] bval = new double[y.nParticles];
double[] gbpos = new double[y.nDims];
double gbval = Double.POSITIVE_INFINITY;
for (int j = 0; j < y.nParticles; ++j) {
// evaluate
v[j] = fn.apply(y.pos[j]);
// update
if (v[j] < y.bval[j]) {
bpos[j] = y.pos[j];
bval[j] = v[j];
} else {
bpos[j] = y.bpos[j];
bval[j] = y.bval[j];
}
if (bval[j] < gbval) {
gbval = bval[j];
gbpos = bpos[j];
}
}
double rg = r.nextDouble();
double[][] pos = new double[y.nParticles][y.nDims];
double[][] vel = new double[y.nParticles][y.nDims];
for (int j = 0; j < y.nParticles; ++j) {
// migrate
double rp = r.nextDouble();
boolean ok = true;
Arrays.fill(vel[j], 0.0);
Arrays.fill(pos[j], 0.0);
for (int k = 0; k < y.nDims; ++k) {
vel[j][k] = p.omega * y.vel[j][k] +
p.phip * rp * (bpos[j][k] - y.pos[j][k]) +
p.phig * rg * (gbpos[k] - y.pos[j][k]);
pos[j][k] = y.pos[j][k] + vel[j][k];
ok = ok && y.min[k] < pos[j][k] && y.max[k] > pos[j][k];
}
if (!ok) {
for (int k = 0; k < y.nDims; ++k) {
pos[j][k] = y.min[k] + (y.max[k] - y.min[k]) * r.nextDouble();
}
}
}
int iter = 1 + y.iter;
return new State(
iter, gbpos, gbval, y.min, y.max, y.parameters,
pos, vel, bpos, bval, y.nParticles, y.nDims
);
}
 
private static State iterate(Function<double[], Double> fn, int n, State y) {
State r = y;
if (n == Integer.MAX_VALUE) {
State old = y;
while (true) {
r = pso(fn, r);
if (Objects.equals(r, old)) break;
old = r;
}
} else {
for (int i = 0; i < n; ++i) {
r = pso(fn, r);
}
}
return r;
}
 
private static double mccormick(double[] x) {
double a = x[0];
double b = x[1];
return Math.sin(a + b) + (a - b) * (a - b) + 1.0 + 2.5 * b - 1.5 * a;
}
 
private static double michalewicz(double[] x) {
int m = 10;
int d = x.length;
double sum = 0.0;
for (int i = 1; i < d; ++i) {
double j = x[i - 1];
double k = Math.sin(i * j * j / Math.PI);
sum += Math.sin(j) * Math.pow(k, 2.0 * m);
}
return -sum;
}
 
public static void main(String[] args) {
State state = psoInit(
new double[]{-1.5, -3.0},
new double[]{4.0, 4.0},
new Parameters(0.0, 0.6, 0.3),
100
);
state = iterate(App::mccormick, 40, state);
state.report("McCormick");
System.out.printf("f(-.54719, -1.54719) : %.15f\n", mccormick(new double[]{-.54719, -1.54719}));
System.out.println();
 
state = psoInit(
new double[]{0.0, 0.0},
new double[]{Math.PI, Math.PI},
new Parameters(0.3, 3.0, 0.3),
1000
);
state = iterate(App::michalewicz, 30, state);
state.report("Michalewicz (2D)");
System.out.printf("f(2.20, 1.57)  : %.15f\n", michalewicz(new double[]{2.20, 1.57}));
}
}
Output:
Test Function        : McCormick
Iterations           : 40
Global Best Position : [-0.5468738679864172, -1.547048532862534]
Global Best value    : -1.913222827709136
f(-.54719, -1.54719) : -1.913222954882274

Test Function        : Michalewicz (2D)
Iterations           : 30
Global Best Position : [2.2029055320517994, 1.832848319327826]
Global Best value    : -0.801303410098550
f(2.20, 1.57)        : -0.801166387820286

Kotlin[edit]

Translation of: JavaScript
// version 1.1.51
 
import java.util.Random
 
typealias Func = (DoubleArray) -> Double
 
class Parameters(val omega: Double, val phip: Double, val phig: Double)
 
class State(
val iter: Int,
val gbpos: DoubleArray,
val gbval: Double,
val min: DoubleArray,
val max: DoubleArray,
val parameters: Parameters,
val pos: Array<DoubleArray>,
val vel: Array<DoubleArray>,
val bpos: Array<DoubleArray>,
val bval: DoubleArray,
val nParticles: Int,
val nDims: Int
) {
fun report(testfunc: String) {
println("Test Function  : $testfunc")
println("Iterations  : $iter")
println("Global Best Position : ${gbpos.asList()}")
println("Global Best Value  : $gbval")
}
}
 
fun psoInit(
min: DoubleArray,
max: DoubleArray,
parameters: Parameters,
nParticles: Int
): State {
val nDims = min.size
val pos = Array(nParticles) { min }
val vel = Array(nParticles) { DoubleArray(nDims) }
val bpos = Array(nParticles) { min }
val bval = DoubleArray(nParticles) { Double.POSITIVE_INFINITY}
val iter = 0
val gbpos = DoubleArray(nDims) { Double.POSITIVE_INFINITY }
val gbval = Double.POSITIVE_INFINITY
return State(iter, gbpos, gbval, min, max, parameters,
pos, vel, bpos, bval, nParticles, nDims)
}
 
val r = Random()
 
fun pso(fn: Func, y: State): State {
val p = y.parameters
val v = DoubleArray(y.nParticles)
val bpos = Array(y.nParticles) { y.min }
val bval = DoubleArray(y.nParticles)
var gbpos = DoubleArray(y.nDims)
var gbval = Double.POSITIVE_INFINITY
for (j in 0 until y.nParticles) {
// evaluate
v[j] = fn(y.pos[j])
// update
if (v[j] < y.bval[j]) {
bpos[j] = y.pos[j]
bval[j] = v[j]
}
else {
bpos[j] = y.bpos[j]
bval[j] = y.bval[j]
}
if (bval[j] < gbval) {
gbval = bval[j]
gbpos = bpos[j]
}
}
val rg = r.nextDouble()
val pos = Array(y.nParticles) { DoubleArray(y.nDims) }
val vel = Array(y.nParticles) { DoubleArray(y.nDims) }
for (j in 0 until y.nParticles) {
// migrate
val rp = r.nextDouble()
var ok = true
vel[j].fill(0.0)
pos[j].fill(0.0)
for (k in 0 until y.nDims) {
vel[j][k] = p.omega * y.vel[j][k] +
p.phip * rp * (bpos[j][k] - y.pos[j][k]) +
p.phig * rg * (gbpos[k] - y.pos[j][k])
pos[j][k] = y.pos[j][k] + vel[j][k]
ok = ok && y.min[k] < pos[j][k] && y.max[k] > pos[j][k]
}
if (!ok) {
for (k in 0 until y.nDims) {
pos[j][k]= y.min[k] + (y.max[k] - y.min[k]) * r.nextDouble()
}
}
}
val iter = 1 + y.iter
return State(
iter, gbpos, gbval, y.min, y.max, y.parameters,
pos, vel, bpos, bval, y.nParticles, y.nDims
)
}
 
fun iterate(fn: Func, n: Int, y: State): State {
var r = y
var old = y
if (n == Int.MAX_VALUE) {
while (true) {
r = pso(fn, r)
if (r == old) break
old = r
}
}
else {
repeat(n) { r = pso(fn, r) }
}
return r
}
 
fun mccormick(x: DoubleArray): Double {
val (a, b) = x
return Math.sin(a + b) + (a - b) * (a - b) + 1.0 + 2.5 * b - 1.5 * a
}
 
fun michalewicz(x: DoubleArray): Double {
val m = 10
val d = x.size
var sum = 0.0
for (i in 1..d) {
val j = x[i - 1]
val k = Math.sin(i * j * j / Math.PI)
sum += Math.sin(j) * Math.pow(k, 2.0 * m)
}
return -sum
}
 
fun main(args: Array<String>) {
var state = psoInit(
min = doubleArrayOf(-1.5, -3.0),
max = doubleArrayOf(4.0, 4.0),
parameters = Parameters(0.0, 0.6, 0.3),
nParticles = 100
)
state = iterate(::mccormick, 40, state)
state.report("McCormick")
println("f(-.54719, -1.54719) : ${mccormick(doubleArrayOf(-.54719, -1.54719))}")
println()
state = psoInit(
min = doubleArrayOf(0.0, 0.0),
max = doubleArrayOf(Math.PI, Math.PI),
parameters = Parameters(0.3, 0.3, 0.3),
nParticles = 1000
)
state = iterate(::michalewicz, 30, state)
state.report("Michalewicz (2D)")
println("f(2.20, 1.57)  : ${michalewicz(doubleArrayOf(2.2, 1.57))}")
}

Sample output:

Test Function        : McCormick
Iterations           : 40
Global Best Position : [-0.5471015946082899, -1.5471991634200966]
Global Best Value    : -1.913222941607108
f(-.54719, -1.54719) : -1.913222954882274

Test Function        : Michalewicz (2D)
Iterations           : 30
Global Best Position : [2.202908690715102, 1.5707970450218895]
Global Best Value    : -1.8013034099142804
f(2.20, 1.57)        : -1.801140718473825

ooRexx[edit]

/* 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
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

Phix[edit]

Translation of: Kotlin
enum OMEGA, PHIP, PHIG
enum ITER,GBPOS,GBVAL,MIN,MAX,PARAMS,POS,VEL,BPOS,BVAL,NPARTICLES,NDIMS
 
constant inf = 1e308*1e308
 
constant fmt = """
Test Function  : %s
Iterations  : %d
Global Best Position : %s
Global Best Value  : %f
"""
 
procedure report(sequence state, string testfunc)
printf(1,fmt,{testfunc,state[ITER],sprint(state[GBPOS]),state[GBVAL]})
end procedure
 
function psoInit(sequence mins, maxs, params, integer nParticles)
integer nDims = length(mins), iter=0
atom gbval = inf
sequence gbpos = repeat(inf,nDims),
pos = repeat(mins,nParticles),
vel = repeat(repeat(0,nDims),nParticles),
bpos = repeat(mins,nParticles),
bval = repeat(inf,nParticles)
return {iter,gbpos,gbval,mins,maxs,params,pos,vel,bpos,bval,nParticles,nDims}
end function
 
function pso(integer fn, sequence state)
integer particles = state[NPARTICLES],
dims = state[NDIMS]
sequence p = state[PARAMS],
v = repeat(0,particles),
bpos = repeat(state[MIN],particles),
bval = repeat(0,particles),
gbpos = repeat(0,dims)
atom gbval = inf
for j=1 to particles do
-- evaluate
v[j] = call_func(fn,{state[POS][j]})
-- update
if v[j] < state[BVAL][j] then
bpos[j] = state[POS][j]
bval[j] = v[j]
else
bpos[j] = state[BPOS][j]
bval[j] = state[BVAL][j]
end if
if bval[j] < gbval then
gbval = bval[j]
gbpos = bpos[j]
end if
end for
atom rg = rnd()
sequence pos = repeat(repeat(0,dims),particles),
vel = repeat(repeat(0,dims),particles)
for j=1 to particles do
-- migrate
atom rp = rnd()
bool ok = true
vel[j] = repeat(0,dims)
pos[j] = repeat(0,dims)
for k=1 to dims do
vel[j][k] = p[OMEGA] * state[VEL][j][k] +
p[PHIP] * rp * (bpos[j][k] - state[POS][j][k]) +
p[PHIG] * rg * (gbpos[k] - state[POS][j][k])
pos[j][k] = state[POS][j][k] + vel[j][k]
ok = ok and state[MIN][k] < pos[j][k] and state[MAX][k] > pos[j][k]
end for
if not ok then
for k=1 to dims do
pos[j][k]= state[MIN][k] + (state[MAX][k] - state[MIN][k]) * rnd()
end for
end if
end for
integer iter = 1 + state[ITER]
return {iter, gbpos, gbval, state[MIN], state[MAX], state[PARAMS],
pos, vel, bpos, bval, particles, dims}
end function
 
function iterate(integer fn, n, sequence state)
sequence r = state,
old = state
if n=-1 then
while true do
r = pso(fn, r)
if (r == old) then exit end if
old = r
end while
else
for i=1 to n do
r = pso(fn, r)
end for
end if
return r
end function
 
function mccormick(sequence x)
atom {a, b} = x
return sin(a + b) + (a - b) * (a - b) + 1.0 + 2.5 * b - 1.5 * a
end function
constant r_mccormick = routine_id("mccormick")
 
function michalewicz(sequence x)
integer m = 10,
d = length(x)
atom total = 0.0
for i=1 to d do
atom j = x[i],
k = sin(i * j * j / PI)
total += sin(j) * power(k, 2.0 * m)
end for
return -total
end function
constant r_michalewicz = routine_id("michalewicz")
 
procedure main()
sequence mins = {-1.5, -3.0},
maxs = {4.0, 4.0},
params = {0.0, 0.6, 0.3}
integer nParticles = 100
sequence state = psoInit(mins,maxs,params,nParticles)
state = iterate(r_mccormick, 40, state)
report(state,"McCormick")
atom {x,y} = state[GBPOS]
printf(1,"f(%.4f, %.4f)  : %f\n\n",{x,y,mccormick({x,y})})
 
mins = {0.0, 0.0}
maxs = {PI, PI}
params = {0.3, 0.3, 0.3}
nParticles = 1000
state = psoInit(mins,maxs,params,nParticles)
state = iterate(r_michalewicz, 30, state)
report(state,"Michalewicz (2D)")
{x,y} = state[GBPOS]
printf(1,"f(%.5f, %.5f)  : %f\n\n",{x,y,michalewicz({x,y})})
end procedure
main()
Output:
Test Function        : McCormick
Iterations           : 40
Global Best Position : {-0.5471808566,-1.547021879}
Global Best Value    : -1.913223
f(-0.5472, -1.5470)  : -1.913223

Test Function        : Michalewicz (2D)
Iterations           : 30
Global Best Position : {2.202905614,1.570796293}
Global Best Value    : -1.801303
f(2.20291, 1.57080)  : -1.801303

Racket[edit]

#lang racket/base
(require racket/list racket/math)
 
(define (unbox-into-cycle s)
(if (box? s) (in-cycle (in-value (unbox s))) s))
 
;; Tries to "maximise" function > (so if you want a minimum, set #:> to <, IYSWIM)
(define (PSO f particles iterations hi lo #:ω ω #:φ_p φ_p #:φ_g φ_g #:> (>? >))
(define dimensions (procedure-arity f))
(unless (exact-nonnegative-integer? dimensions)
(raise-argument-error 'PSO "function of fixed arity" 1 f))
 
(define-values (x v)
(for/lists (x v)
((_ particles))
(for/lists (xi vi)
((d (in-range dimensions))
(h (unbox-into-cycle hi))
(l (unbox-into-cycle lo)))
(define h-l (- h l))
(values (+ l (* (random) h-l)) (+ (- h-l) (* 2 (random) h-l))))))
 
(define (particle-step x_i v_i p_i g)
(for/lists (x_i+ v_i+)
((x_id (in-list x_i))
(v_id (in-list v_i))
(p_id (in-list p_i))
(g_d (in-list g)))
(define v_id+ (+ (* ω v_id)
(* φ_p (random) (- p_id x_id))
(* φ_g (random) (- g_d x_id))))
(values (+ x_id v_id+) v_id+)))
 
(define (call-f args) (apply f args))
(define g0 (argmax call-f x))
(define-values (_X _V _P _P. G G.)
(for/fold ; because of g and g., we can't use for/lists
((X x) (V v) (P x) (P. (map call-f x)) (g g0) (g. (apply f g0)))
((_ iterations))
(for/fold
((x+ null) (v+ null) (p+ null) (p.+ null) (g+ g) (g.+ g.))
((x_i (in-list X))
(v_i (in-list V))
(p_i (in-list P))
(p._i (in-list P.)))
(define-values (x_i+ v_i+) (particle-step x_i v_i p_i g+))
(let* ((x._i+ (apply f x_i+))
(new-p_i? (>? x._i+ p._i))
(new-g? (>? x._i+ g.+)))
(values (cons x_i+ x+)
(cons v_i+ v+)
(cons (if new-p_i? x_i+ p_i) p+)
(cons (if new-p_i? x._i+ p._i) p.+)
(if new-g? x_i+ g+)
(if new-g? x._i+ g.+))))))
(values G G.))
 
(define (McCormick x1 x2)
(+ (sin (+ x1 x2)) (sqr (- x1 x2)) (* -1.5 x1) (* 2.5 x2) 1))
 
(define (Michalewitz d #:m (m 10))
(define 2m (* 2 m))
(define /pi (/ pi))
(define (f . xx)
(let Σ ((s 0) (i 1) (xx xx))
(if (null? xx)
(- s)
(let ((x (car xx)))
(Σ (+ s (* (sin x) (expt (sin (* i (sqr x) /pi)) 2m))) (+ i 1) (cdr xx))))))
(procedure-reduce-arity f d))
 
(displayln "McCormick [-1.993] @ (-0.54719, -1.54719)")
(PSO McCormick 1000 100 #(-1.5 -3) #(4 4) #:ω 0 #:φ_p 0.6 #:φ_g 0.3 #:> <)
(displayln "Michalewitz 2d [-1.8013] @ (2.20, 1.57)")
(PSO (Michalewitz 2) 1000 30 (box 0) (box pi) #:ω 0.3 #:φ_p 0.3 #:φ_g 0.3 #:> <)
(displayln "Michalewitz 5d [-4.687658]")
(PSO (Michalewitz 5) 1000 30 (box 0) (box pi) #:ω 0.3 #:φ_p 0.3 #:φ_g 0.3 #:> <)
(displayln "Michalewitz 10d [-9.66015]")
(PSO (Michalewitz 10) 1000 30 (box 0) (box pi) #:ω 0.3 #:φ_p 0.3 #:φ_g 0.3 #:> <)
Output:

Here is a sample run, the particles roll downhill quite nicely for McCormick, but there's a lot of space to search with the 10-dimensional Michalewitz; so YMMV with that one!

McCormick [-1.993] @ (-0.54719, -1.54719)
'(-0.5471975539492846 -1.547197548223612)
-1.9132229549810367
Michalewitz 2d [-1.8013] @ (2.20, 1.57)
'(2.20290527060906 1.5707963523178217)
-1.8013034100975123
Michalewitz 5d [-4.687658]
'(2.188617053067511
  1.571283730996248
  1.2884975345181757
  1.9194689579781514
  1.7202092563763838)
-4.680722049442259
Michalewitz 10d [-9.66015]
'(1.359756739301337
  2.7216986742916007
  1.2823734619604734
  1.097509491839529
  2.2225042675789752
  0.9162856379217913
  1.8753760783453128
  0.7909979596555162
  0.46574677476493
  1.8558804696523914)
-6.432092623300999

REXX[edit]

Translation of: ooRexx

This REXX version uses a large   numeric digits   (the number of decimal digits in pi),   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,   110).

This REXX version supports the specifying of   X,   Y,   and   D,   as well as the number of particles,   and the number of
decimal digits to be displayed.   A little extra code was added to show a title and align the output columns.

The refinement loop is stopped when the calculation of the function value stabilizes.

Note that REXX uses decimal floating point, not binary.

/*REXX program calculates Particle Swarm Optimization as it migrates through a solution.*/
numeric digits length( pi() ) - 1 /*use the number of decimal digs in pi.*/
parse arg x y d #part sDigs . /*obtain optional arguments from the CL*/
if x=='' | x=="," then x= -0.5 /*Not specified? Then use the default.*/
if y=='' | y=="," then y= -1.5 /* " " " " " " */
if d=='' | d=="," then d= 1 /* " " " " " " */
if #part=='' | #part=="," then #part= 1e12 /* " " " " " " */
if sDigs=='' | sDigs=="," then sDigs= 25 /* " " " " " " */
old= /*#part: 1e12 ≡ is one trillion. */
minF= #part /*the minimum for the function (#part).*/
show= sDigs + 3 /*adjust number decimal digits for show*/
say "══iteration══" center('X',show,"═") center('Y',show,"═") center('D',show,"═")
#= 0 /*the number of iterations for REFINE.*/
call refine x,y /* [↓] same as ÷ by five.*/
do until refine(minX, minY) /*perform until the mix is "refined". */
d=d * .2 /*decrease the difference in the mix. .*/
end /*until*/ /* [↑] stop refining if no difference.*/
say
$= 15 + show * 2 /*compute the indentation for alignment*/
say right('The global minimum for f(-.54719, -1.54719) ───► ', $) fmt(f(-.54719, -1.54719))
say right('The published global minimum is:' , $) fmt( -1.9133 )
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
refine: parse arg xx,yy; h= d * .5 /*compute ½ distance.*/
do x=xx-d to xx+d by h
do y=yy-d to yy+d by h; f= f(x, y); if f>=minF then iterate
new= fmt(x) fmt(y) fmt(f); if new=old then return 1
#= # + 1 /*bump # iterations. */
say center(#,13) new /*show " " */
minF= f; minX= x; minY= y; old= new /*assign new values. */
end /*y*/
end /*x*/
return 0
/*──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────*/
f: procedure: parse arg a,b; return sin(a+b) + (a-b)**2 - 1.5*a + 2.5*b + 1
fmt: ?=format(arg(1), , sDigs); L=length(?); if pos(., ?)\==0 then ?=strip( strip(?, 'T', 0), "T", .); return left(?, L)
pi: pi=3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865; return pi
r2r: return arg(1) // ( pi() * 2) /*normalize radians ───► a unit circle.*/
sin: procedure; arg x; x= r2r(x); z= x; xx=x*x; do k=2 by 2 until p=z; p=z; x= -x*xx/ (k*(k+1)); z= z+x; end; return z
output   when using the default input:
══iteration══ ═════════════X══════════════ ═════════════Y══════════════ ═════════════D══════════════
      1       -1.5                         -2.5                         -1.2431975046920717486273609
      2       -1                           -2                           -1.6411200080598672221007448
      3       -0.5                         -1.5                         -1.9092974268256816953960199
      4       -0.54                        -1.54                        -1.9131329795075164948766768
      5       -0.548                       -1.548                       -1.9132218400165267634506035
      6       -0.548                       -1.5472                      -1.9132220344928294065568196
      7       -0.5472                      -1.5472                      -1.9132229549706499208388746
      8       -0.5472                      -1.54719872                  -1.9132229549737311254290577
      9       -0.54719872                  -1.54719872                  -1.9132229549786702369612333
     10       -0.54719872                  -1.54719744                  -1.91322295497891365438682
     11       -0.54719744                  -1.54719744                  -1.9132229549810149766572388
     12       -0.5471975424                -1.5471975424                -1.9132229549810362588916172
     13       -0.54719755264               -1.54719755264               -1.9132229549810363893093655
     14       -0.547197550592              -1.547197550592              -1.9132229549810363922848065
     15       -0.5471975514112             -1.5471975514112             -1.9132229549810363928381695
     16       -0.5471975510016             -1.5471975510016             -1.9132229549810363928520779
     17       -0.54719755116544            -1.54719755116544            -1.9132229549810363929162561
     18       -0.547197551198208           -1.547197551198208           -1.9132229549810363929179331
     19       -0.547197551198208           -1.54719755119755264         -1.9132229549810363929179344
     20       -0.54719755119755264         -1.54719755119755264         -1.9132229549810363929179361
     21       -0.54719755119755264         -1.54719755119689728         -1.9132229549810363929179365
     22       -0.54719755119689728         -1.54719755119689728         -1.9132229549810363929179375
     23       -0.54719755119689728         -1.547197551196766208        -1.9132229549810363929179375
     24       -0.547197551196766208        -1.547197551196766208        -1.9132229549810363929179376
     25       -0.547197551196766208        -1.547197551196635136        -1.9132229549810363929179376
     26       -0.547197551196635136        -1.547197551196635136        -1.9132229549810363929179376
     27       -0.547197551196635136        -1.5471975511966089216       -1.9132229549810363929179376
     28       -0.5471975511966089216       -1.5471975511966089216       -1.9132229549810363929179376
     29       -0.5471975511966089216       -1.54719755119660367872      -1.9132229549810363929179376
     30       -0.54719755119660367872      -1.54719755119660367872      -1.9132229549810363929179376
     31       -0.54719755119660367872      -1.54719755119659843584      -1.9132229549810363929179376
     32       -0.54719755119659843584      -1.54719755119659843584      -1.9132229549810363929179376
     33       -0.547197551196597387264     -1.547197551196597387264     -1.9132229549810363929179376
     34       -0.5471975511965978066944    -1.5471975511965978066944    -1.9132229549810363929179376
     35       -0.5471975511965978066944    -1.54719755119659776475136   -1.9132229549810363929179376
     36       -0.54719755119659776475136   -1.54719755119659776475136   -1.9132229549810363929179376
     37       -0.54719755119659776475136   -1.547197551196597756362752  -1.9132229549810363929179376
     38       -0.547197551196597756362752  -1.547197551196597756362752  -1.9132229549810363929179376
     39       -0.547197551196597756362752  -1.547197551196597747974144  -1.9132229549810363929179376
     40       -0.547197551196597747974144  -1.547197551196597747974144  -1.9132229549810363929179376
     41       -0.547197551196597747974144  -1.5471975511965977462964224 -1.9132229549810363929179376
     42       -0.5471975511965977462964224 -1.5471975511965977462964224 -1.9132229549810363929179376
     43       -0.5471975511965977462964224 -1.5471975511965977462293135 -1.9132229549810363929179376
     44       -0.5471975511965977462293135 -1.5471975511965977462293135 -1.9132229549810363929179376
     45       -0.5471975511965977462293135 -1.5471975511965977461622047 -1.9132229549810363929179376
     46       -0.5471975511965977461622047 -1.5471975511965977461622047 -1.9132229549810363929179376
     47       -0.5471975511965977461487829 -1.5471975511965977461487829 -1.9132229549810363929179376
     48       -0.5471975511965977461541516 -1.5471975511965977461541516 -1.9132229549810363929179376
     49       -0.547197551196597746154259  -1.547197551196597746154259  -1.9132229549810363929179376
     50       -0.547197551196597746154259  -1.5471975511965977461542375 -1.9132229549810363929179376
     51       -0.5471975511965977461542375 -1.5471975511965977461542375 -1.9132229549810363929179376
     52       -0.5471975511965977461542375 -1.547197551196597746154216  -1.9132229549810363929179376
     53       -0.547197551196597746154216  -1.547197551196597746154216  -1.9132229549810363929179376
     54       -0.547197551196597746154216  -1.5471975511965977461542152 -1.9132229549810363929179376
     55       -0.5471975511965977461542152 -1.5471975511965977461542152 -1.9132229549810363929179376
     56       -0.5471975511965977461542152 -1.5471975511965977461542143 -1.9132229549810363929179376
     57       -0.5471975511965977461542143 -1.5471975511965977461542143 -1.9132229549810363929179376
     58       -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