Minimizing with differential evolutionMinimizing 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

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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













4












$begingroup$


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










share|improve this question











$endgroup$











  • $begingroup$
    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
    56 mins ago
















4












$begingroup$


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










share|improve this question











$endgroup$











  • $begingroup$
    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
    56 mins ago














4












4








4


2



$begingroup$


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










share|improve this question











$endgroup$




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







mathematical-optimization






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited 1 hour ago







Okkes Dulgerci

















asked 3 hours ago









Okkes DulgerciOkkes Dulgerci

5,2691917




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
    56 mins ago

















  • $begingroup$
    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
    56 mins ago
















$begingroup$
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
56 mins ago





$begingroup$
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
56 mins ago











1 Answer
1






active

oldest

votes


















3












$begingroup$

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.






share|improve this answer









$endgroup$












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    1 Answer
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    active

    oldest

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    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    3












    $begingroup$

    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.






    share|improve this answer









    $endgroup$

















      3












      $begingroup$

      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.






      share|improve this answer









      $endgroup$















        3












        3








        3





        $begingroup$

        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.






        share|improve this answer









        $endgroup$



        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.







        share|improve this answer












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        answered 2 hours ago









        Michael E2Michael E2

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        148k12198478



























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