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Minimizing with differential evolution

Minimizing a function of many coordinatesMinimizing a function with some restrictionsMinimizing Multiple FunctionsProblem in minimizing expressionSolving 4 coupled differential equation and minimizing the solutionProblem when minimizing user-defined function in Mathematica with Minimize[]Minimizing with constraintsMinimizing functions with parametersMinimizing a conditional function with parametersMinimizing a function problem

A differential evolution algorithm is given here. I would like to get this kind of animation. I thought I could use NMinimize, given
DifferentialEvolution as an option, but it turns out that does not work as I espected.

Is it possible to extract intermediate step in DifferentialEvolution, or do I have to implement algorithm myself?

f[x_, y_] := 

  -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - E^(0.5 (Cos[2 π x] + Cos[2 π y])) + E + 20



p1 = 

  Plot3D[f[x, y], {x, -5, 5}, {y, -5, 5}, 

  PerformanceGoal -> "Quality",

  ColorFunction -> "WatermelonColors", 

  Mesh -> None, 

  BoxRatios -> {1, 1, 1}];   



p2 = 

   DensityPlot[f[x, y], {x, -5, 5}, {y, -5, 5}, 

     ColorFunction -> "WatermelonColors", 

     PlotPoints -> 200, 

     PerformanceGoal -> "Quality", 

     Frame -> False, 

     PlotRangePadding -> None];



p3 = Plot3D[0, {x, -5, 5}, {y, -5, 5}, PlotStyle -> Texture[p2], Mesh -> None];



Show[p1, p3, PlotRange -> {0, 15}]

enter image description here

When I use StepMonitor to track iterations as follows, it does not work.

{fit, intermediates} = 

  Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y}, 

    MaxIterations -> 1000, 

    Method -> {"DifferentialEvolution", "InitialPoints" -> Tuples[Range[-5, 5], 2]}, 

    StepMonitor :> Sow[{x, y}]]];



Table[

  ListPlot[Take[intermediates[[1, i ;; i + 10]]], 

    Frame -> True, ImageSize -> 350, AspectRatio -> 1], 

  {i, 10, 1000, 100}]

EDIT
Here is the result when we used @Michael E2 solution. Cool!!

f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - 

  E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20



p1 = Plot3D[f[x, y], {x, -5, 5}, {y, -5, 5}, 

   PerformanceGoal -> "Quality", ColorFunction -> "WatermelonColors", 

   Mesh -> None, BoxRatios -> {1, 1, 1}];



p2 = DensityPlot[f[x, y], {x, -5, 5}, {y, -5, 5}, 

   ColorFunction -> "WatermelonColors", PerformanceGoal -> "Quality", 

   Frame -> False, PlotRangePadding -> None];



p3 = Plot3D[-0.5, {x, -5, 5}, {y, -5, 5}, PlotStyle -> Texture[p2], 

   Mesh -> None];



p4 = Show[p1, p3, PlotRange -> {-0.5, 15}]



Block[{f}, 

  f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - 

    E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20;

  {fit, intermediates} = 

   Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y}, 

     MaxIterations -> 30, 

     Method -> {"DifferentialEvolution", 

       "InitialPoints" -> Tuples[Range[-5, 5], 2]}, 

     StepMonitor :> 

      Sow[{Optimization`NMinimizeDump`vecs, 

        Optimization`NMinimizeDump`vals}]]];] // Quiet



Multicolumn[

 Table[Show[p4, 

   ListPointPlot3D[{Append[#, 0] & /@ intermediates[[1, i, 1]]}, 

    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, Boxed -> False, 

    PlotStyle -> Directive[AbsolutePointSize[3], Black]]], {i, 1, 30, 

   2}], 5, Appearance -> "Horizontal"]

enter image description here

edited 1 hour ago

asked 2 hours ago

Okkes Dulgerci

5,2691917

$begingroup$
Note that blocking f (Block[{f}, ...]) isn't necessary. It was just to prevent f from being defined, which is a habit I have with single-lettter symbols on SE, esp. ones I use like f, x, etc. -- thanks for the accept!
$endgroup$
– Michael E2
46 mins ago

add a comment |

Is it possible to extract intermediate step in DifferentialEvolution, or do I have to implement algorithm myself?

f[x_, y_] := 

  -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - E^(0.5 (Cos[2 π x] + Cos[2 π y])) + E + 20



p1 = 

  Plot3D[f[x, y], {x, -5, 5}, {y, -5, 5}, 

  PerformanceGoal -> "Quality",

  ColorFunction -> "WatermelonColors", 

  Mesh -> None, 

  BoxRatios -> {1, 1, 1}];   



p2 = 

   DensityPlot[f[x, y], {x, -5, 5}, {y, -5, 5}, 

     ColorFunction -> "WatermelonColors", 

     PlotPoints -> 200, 

     PerformanceGoal -> "Quality", 

     Frame -> False, 

     PlotRangePadding -> None];



p3 = Plot3D[0, {x, -5, 5}, {y, -5, 5}, PlotStyle -> Texture[p2], Mesh -> None];



Show[p1, p3, PlotRange -> {0, 15}]

enter image description here

When I use StepMonitor to track iterations as follows, it does not work.

{fit, intermediates} = 

  Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y}, 

    MaxIterations -> 1000, 

    Method -> {"DifferentialEvolution", "InitialPoints" -> Tuples[Range[-5, 5], 2]}, 

    StepMonitor :> Sow[{x, y}]]];



Table[

  ListPlot[Take[intermediates[[1, i ;; i + 10]]], 

    Frame -> True, ImageSize -> 350, AspectRatio -> 1], 

  {i, 10, 1000, 100}]

EDIT
Here is the result when we used @Michael E2 solution. Cool!!

f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - 

  E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20



p1 = Plot3D[f[x, y], {x, -5, 5}, {y, -5, 5}, 

   PerformanceGoal -> "Quality", ColorFunction -> "WatermelonColors", 

   Mesh -> None, BoxRatios -> {1, 1, 1}];



p2 = DensityPlot[f[x, y], {x, -5, 5}, {y, -5, 5}, 

   ColorFunction -> "WatermelonColors", PerformanceGoal -> "Quality", 

   Frame -> False, PlotRangePadding -> None];



p3 = Plot3D[-0.5, {x, -5, 5}, {y, -5, 5}, PlotStyle -> Texture[p2], 

   Mesh -> None];



p4 = Show[p1, p3, PlotRange -> {-0.5, 15}]



Block[{f}, 

  f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - 

    E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20;

  {fit, intermediates} = 

   Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y}, 

     MaxIterations -> 30, 

     Method -> {"DifferentialEvolution", 

       "InitialPoints" -> Tuples[Range[-5, 5], 2]}, 

     StepMonitor :> 

      Sow[{Optimization`NMinimizeDump`vecs, 

        Optimization`NMinimizeDump`vals}]]];] // Quiet



Multicolumn[

 Table[Show[p4, 

   ListPointPlot3D[{Append[#, 0] & /@ intermediates[[1, i, 1]]}, 

    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, Boxed -> False, 

    PlotStyle -> Directive[AbsolutePointSize[3], Black]]], {i, 1, 30, 

   2}], 5, Appearance -> "Horizontal"]

enter image description here

edited 1 hour ago

asked 2 hours ago

Okkes Dulgerci

5,2691917

$begingroup$
Note that blocking f (Block[{f}, ...]) isn't necessary. It was just to prevent f from being defined, which is a habit I have with single-lettter symbols on SE, esp. ones I use like f, x, etc. -- thanks for the accept!
$endgroup$
– Michael E2
46 mins ago

add a comment |

Is it possible to extract intermediate step in DifferentialEvolution, or do I have to implement algorithm myself?

f[x_, y_] := 

  -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - E^(0.5 (Cos[2 π x] + Cos[2 π y])) + E + 20



p1 = 

  Plot3D[f[x, y], {x, -5, 5}, {y, -5, 5}, 

  PerformanceGoal -> "Quality",

  ColorFunction -> "WatermelonColors", 

  Mesh -> None, 

  BoxRatios -> {1, 1, 1}];   



p2 = 

   DensityPlot[f[x, y], {x, -5, 5}, {y, -5, 5}, 

     ColorFunction -> "WatermelonColors", 

     PlotPoints -> 200, 

     PerformanceGoal -> "Quality", 

     Frame -> False, 

     PlotRangePadding -> None];



p3 = Plot3D[0, {x, -5, 5}, {y, -5, 5}, PlotStyle -> Texture[p2], Mesh -> None];



Show[p1, p3, PlotRange -> {0, 15}]

enter image description here

When I use StepMonitor to track iterations as follows, it does not work.

{fit, intermediates} = 

  Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y}, 

    MaxIterations -> 1000, 

    Method -> {"DifferentialEvolution", "InitialPoints" -> Tuples[Range[-5, 5], 2]}, 

    StepMonitor :> Sow[{x, y}]]];



Table[

  ListPlot[Take[intermediates[[1, i ;; i + 10]]], 

    Frame -> True, ImageSize -> 350, AspectRatio -> 1], 

  {i, 10, 1000, 100}]

EDIT
Here is the result when we used @Michael E2 solution. Cool!!

f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - 

  E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20



p1 = Plot3D[f[x, y], {x, -5, 5}, {y, -5, 5}, 

   PerformanceGoal -> "Quality", ColorFunction -> "WatermelonColors", 

   Mesh -> None, BoxRatios -> {1, 1, 1}];



p2 = DensityPlot[f[x, y], {x, -5, 5}, {y, -5, 5}, 

   ColorFunction -> "WatermelonColors", PerformanceGoal -> "Quality", 

   Frame -> False, PlotRangePadding -> None];



p3 = Plot3D[-0.5, {x, -5, 5}, {y, -5, 5}, PlotStyle -> Texture[p2], 

   Mesh -> None];



p4 = Show[p1, p3, PlotRange -> {-0.5, 15}]



Block[{f}, 

  f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - 

    E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20;

  {fit, intermediates} = 

   Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y}, 

     MaxIterations -> 30, 

     Method -> {"DifferentialEvolution", 

       "InitialPoints" -> Tuples[Range[-5, 5], 2]}, 

     StepMonitor :> 

      Sow[{Optimization`NMinimizeDump`vecs, 

        Optimization`NMinimizeDump`vals}]]];] // Quiet



Multicolumn[

 Table[Show[p4, 

   ListPointPlot3D[{Append[#, 0] & /@ intermediates[[1, i, 1]]}, 

    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, Boxed -> False, 

    PlotStyle -> Directive[AbsolutePointSize[3], Black]]], {i, 1, 30, 

   2}], 5, Appearance -> "Horizontal"]

enter image description here

edited 1 hour ago

asked 2 hours ago

Okkes Dulgerci

5,2691917

Is it possible to extract intermediate step in DifferentialEvolution, or do I have to implement algorithm myself?

f[x_, y_] := 

  -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - E^(0.5 (Cos[2 π x] + Cos[2 π y])) + E + 20



p1 = 

  Plot3D[f[x, y], {x, -5, 5}, {y, -5, 5}, 

  PerformanceGoal -> "Quality",

  ColorFunction -> "WatermelonColors", 

  Mesh -> None, 

  BoxRatios -> {1, 1, 1}];   



p2 = 

   DensityPlot[f[x, y], {x, -5, 5}, {y, -5, 5}, 

     ColorFunction -> "WatermelonColors", 

     PlotPoints -> 200, 

     PerformanceGoal -> "Quality", 

     Frame -> False, 

     PlotRangePadding -> None];



p3 = Plot3D[0, {x, -5, 5}, {y, -5, 5}, PlotStyle -> Texture[p2], Mesh -> None];



Show[p1, p3, PlotRange -> {0, 15}]

enter image description here

When I use StepMonitor to track iterations as follows, it does not work.

{fit, intermediates} = 

  Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y}, 

    MaxIterations -> 1000, 

    Method -> {"DifferentialEvolution", "InitialPoints" -> Tuples[Range[-5, 5], 2]}, 

    StepMonitor :> Sow[{x, y}]]];



Table[

  ListPlot[Take[intermediates[[1, i ;; i + 10]]], 

    Frame -> True, ImageSize -> 350, AspectRatio -> 1], 

  {i, 10, 1000, 100}]

EDIT
Here is the result when we used @Michael E2 solution. Cool!!

f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - 

  E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20



p1 = Plot3D[f[x, y], {x, -5, 5}, {y, -5, 5}, 

   PerformanceGoal -> "Quality", ColorFunction -> "WatermelonColors", 

   Mesh -> None, BoxRatios -> {1, 1, 1}];



p2 = DensityPlot[f[x, y], {x, -5, 5}, {y, -5, 5}, 

   ColorFunction -> "WatermelonColors", PerformanceGoal -> "Quality", 

   Frame -> False, PlotRangePadding -> None];



p3 = Plot3D[-0.5, {x, -5, 5}, {y, -5, 5}, PlotStyle -> Texture[p2], 

   Mesh -> None];



p4 = Show[p1, p3, PlotRange -> {-0.5, 15}]



Block[{f}, 

  f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - 

    E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20;

  {fit, intermediates} = 

   Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y}, 

     MaxIterations -> 30, 

     Method -> {"DifferentialEvolution", 

       "InitialPoints" -> Tuples[Range[-5, 5], 2]}, 

     StepMonitor :> 

      Sow[{Optimization`NMinimizeDump`vecs, 

        Optimization`NMinimizeDump`vals}]]];] // Quiet



Multicolumn[

 Table[Show[p4, 

   ListPointPlot3D[{Append[#, 0] & /@ intermediates[[1, i, 1]]}, 

    PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, Boxed -> False, 

    PlotStyle -> Directive[AbsolutePointSize[3], Black]]], {i, 1, 30, 

   2}], 5, Appearance -> "Horizontal"]

enter image description here

mathematical-optimization

edited 1 hour ago

asked 2 hours ago

Okkes Dulgerci

5,2691917

edited 1 hour ago

asked 2 hours ago

Okkes Dulgerci

5,2691917

edited 1 hour ago

asked 2 hours ago

Okkes Dulgerci

5,2691917

asked 2 hours ago

Okkes Dulgerci

5,2691917

asked 2 hours ago

Okkes Dulgerci

5,2691917

$begingroup$
Note that blocking f (Block[{f}, ...]) isn't necessary. It was just to prevent f from being defined, which is a habit I have with single-lettter symbols on SE, esp. ones I use like f, x, etc. -- thanks for the accept!
$endgroup$
– Michael E2
46 mins ago

add a comment |

$begingroup$
Note that blocking f (Block[{f}, ...]) isn't necessary. It was just to prevent f from being defined, which is a habit I have with single-lettter symbols on SE, esp. ones I use like f, x, etc. -- thanks for the accept!
$endgroup$
– Michael E2
46 mins ago

Note that blocking f (Block[{f}, ...]) isn't necessary. It was just to prevent f from being defined, which is a habit I have with single-lettter symbols on SE, esp. ones I use like f, x, etc. -- thanks for the accept!

– Michael E2
46 mins ago

add a comment |

1 Answer
1

active

oldest

votes

Here's a way:

Block[{f},

 f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - 

   E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20;

 {fit, intermediates} = 

  Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y}, 

    MaxIterations -> 30, 

    Method -> {"DifferentialEvolution", 

      "InitialPoints" -> Tuples[Range[-5, 5], 2]}, 

    StepMonitor :> 

     Sow[{Optimization`NMinimizeDump`vecs, 

       Optimization`NMinimizeDump`vals}]]];

 ]



Manipulate[

 Graphics[{

   PointSize[Medium],

   Point[intermediates[[1, n, 1]], 

    VertexColors -> 

     ColorData["Rainbow"] /@ 

      Rescale[intermediates[[1, n, 2]], 

       MinMax[intermediates[[1, All, 2]]]]]

   },

  PlotRange -> 5, Frame -> True],

 {n, 1, Length@intermediates[[1]], 1}

 ]

enter image description here

You can find out about things like Optimization`NMinimizeDump`vecs by inspecting the code for Optimization`NMinimizeDump`CoreDE.

answered 2 hours ago

Michael E2

148k12198478

add a comment |

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

Here's a way:

Block[{f},

 f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - 

   E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20;

 {fit, intermediates} = 

  Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y}, 

    MaxIterations -> 30, 

    Method -> {"DifferentialEvolution", 

      "InitialPoints" -> Tuples[Range[-5, 5], 2]}, 

    StepMonitor :> 

     Sow[{Optimization`NMinimizeDump`vecs, 

       Optimization`NMinimizeDump`vals}]]];

 ]



Manipulate[

 Graphics[{

   PointSize[Medium],

   Point[intermediates[[1, n, 1]], 

    VertexColors -> 

     ColorData["Rainbow"] /@ 

      Rescale[intermediates[[1, n, 2]], 

       MinMax[intermediates[[1, All, 2]]]]]

   },

  PlotRange -> 5, Frame -> True],

 {n, 1, Length@intermediates[[1]], 1}

 ]

enter image description here

You can find out about things like Optimization`NMinimizeDump`vecs by inspecting the code for Optimization`NMinimizeDump`CoreDE.

answered 2 hours ago

Michael E2

148k12198478

add a comment |

Here's a way:

Block[{f},

 f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - 

   E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20;

 {fit, intermediates} = 

  Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y}, 

    MaxIterations -> 30, 

    Method -> {"DifferentialEvolution", 

      "InitialPoints" -> Tuples[Range[-5, 5], 2]}, 

    StepMonitor :> 

     Sow[{Optimization`NMinimizeDump`vecs, 

       Optimization`NMinimizeDump`vals}]]];

 ]



Manipulate[

 Graphics[{

   PointSize[Medium],

   Point[intermediates[[1, n, 1]], 

    VertexColors -> 

     ColorData["Rainbow"] /@ 

      Rescale[intermediates[[1, n, 2]], 

       MinMax[intermediates[[1, All, 2]]]]]

   },

  PlotRange -> 5, Frame -> True],

 {n, 1, Length@intermediates[[1]], 1}

 ]

enter image description here

You can find out about things like Optimization`NMinimizeDump`vecs by inspecting the code for Optimization`NMinimizeDump`CoreDE.

answered 2 hours ago

Michael E2

148k12198478

add a comment |

Here's a way:

Block[{f},

 f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - 

   E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20;

 {fit, intermediates} = 

  Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y}, 

    MaxIterations -> 30, 

    Method -> {"DifferentialEvolution", 

      "InitialPoints" -> Tuples[Range[-5, 5], 2]}, 

    StepMonitor :> 

     Sow[{Optimization`NMinimizeDump`vecs, 

       Optimization`NMinimizeDump`vals}]]];

 ]



Manipulate[

 Graphics[{

   PointSize[Medium],

   Point[intermediates[[1, n, 1]], 

    VertexColors -> 

     ColorData["Rainbow"] /@ 

      Rescale[intermediates[[1, n, 2]], 

       MinMax[intermediates[[1, All, 2]]]]]

   },

  PlotRange -> 5, Frame -> True],

 {n, 1, Length@intermediates[[1]], 1}

 ]

enter image description here

You can find out about things like Optimization`NMinimizeDump`vecs by inspecting the code for Optimization`NMinimizeDump`CoreDE.

answered 2 hours ago

Michael E2

148k12198478

Here's a way:

Block[{f},

 f[x_, y_] := -20 E^(-0.2 Sqrt[0.5 (x^2 + y^2)]) - 

   E^(0.5 (Cos[2 [Pi] x] + Cos[2 [Pi] y])) + E + 20;

 {fit, intermediates} = 

  Reap[NMinimize[{f[x, y], -5 <= x <= 5, -5 <= y <= 5}, {x, y}, 

    MaxIterations -> 30, 

    Method -> {"DifferentialEvolution", 

      "InitialPoints" -> Tuples[Range[-5, 5], 2]}, 

    StepMonitor :> 

     Sow[{Optimization`NMinimizeDump`vecs, 

       Optimization`NMinimizeDump`vals}]]];

 ]



Manipulate[

 Graphics[{

   PointSize[Medium],

   Point[intermediates[[1, n, 1]], 

    VertexColors -> 

     ColorData["Rainbow"] /@ 

      Rescale[intermediates[[1, n, 2]], 

       MinMax[intermediates[[1, All, 2]]]]]

   },

  PlotRange -> 5, Frame -> True],

 {n, 1, Length@intermediates[[1]], 1}

 ]

enter image description here

You can find out about things like Optimization`NMinimizeDump`vecs by inspecting the code for Optimization`NMinimizeDump`CoreDE.

answered 2 hours ago

Michael E2

148k12198478

answered 2 hours ago

Michael E2

148k12198478

answered 2 hours ago

Michael E2

148k12198478

answered 2 hours ago

Michael E2

148k12198478

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