Skip to main content

mathematical optimization - Simulated Annealing Convergence


I am using Simulated Annealing method for a simulation based optimization of a process that has 3 variables, using NMinimize. I print the input/output during every iteration using the "Evaluation Monitor". I have noticed that after about 3000 iterations, Mathematica gives me a convergence result. But checking the results from "Evaluation Monitor" shows that the ultimate convergence result is not the global minimal, but the process has visited a better minimal during one of the iterations. Why is this happening?


Here is a sample code:



demand[n_,k_]:=Min[k Vf,n capacity];
supply[n_,k_]:=Min[(n Kj-k) w, n capacity];
flo[n_,Ku_,Kd_]:=Min[demand[n,Ku],supply[n,Kd]];
dx=Vf*dt; n=Round[Flen/dx]; m=Round[SimTime/dt];
p=Round[Rlen/dx]; θ=Vf/w; capacity=w*Vf*Kj/(Vf+w); α[a1_]:=1800.; β[a2_]:=0.1; L=3.;
Flen=8.; Rlen=3.; SimTime=30./60.; Kj=150.; w=20.; Vf=100.; dt=12./3600.; d=1.;
RMLocation=Round[(2/3) p]; j=0;

f[a1_,a2_,a3_]:=Module[{b1=a1,b2=a2,b3=a3,TT=0,NtwrkTT=0,j=0},
RM[x_,t_]:=Piecewise[{{100 b1,x<=3},{100 b2,3
NtwrkTT=0; Clear[k0,kr,k,γ];
k0=ConstantArray[0,n];
kr=Table[Table[0,{k,1,p}],{i,1,n}];
γ=ConstantArray[1,n];
For[i=2, i TT=Plus@@(Plus@@kr); NtwrkTT=TT; k=k0;
While[TT>0, TT=0;
For[i=2, i dem=demand[L,k0[[i]]]; dem=If[dem==0,0.001,dem];
γ[[i]]=Min[1,supply[L,k0[[i+1]]]/dem];

ϕ=γ[[i]] demand[1,kr[[i,p]]]/d;
Qr=(ϕ-β[i dx] FQin) dx;
FQout=Min[demand[L,k0[[i]]],supply[L,k0[[i+1]]]];
k[[i]]=k0[[i]]+(FQin-FQout+Qr)/Vf;
kr0=kr[[i]];
For[ir=2,ir<=p,ir++,
MR=If[ir==RMLocation+1,RM[i dx,j dt],capacity];
RQin=Min[MR,If[ir==2,flo[1,kr0[[ir-1]],kr0[[ir]]],RQout]];
MR=If[ir==RMLocation,RM[i dx,j dt],capacity];
RQout=Min[MR,If[ir
kr[[i,ir]]=kr0[[ir]]+(RQin-RQout)/Vf];
kr[[i,1]]=If[j<=m,α[i dx] d/Vf,0]];
TT=Plus@@(Plus@@kr);
TT+=Plus@@k;
k0=k;NtwrkTT+=TT;j++];
NtwrkTT dt]
NMinimize[{f[a,b,c],3<=a<=12&&3<=b<=12&&3<=c<=12&&Element[a|b|c,Integers]},{a,b,c},Method->{"SimulatedAnnealing","SearchPoints"->5^5},EvaluationMonitor:>Print["a = ",a," , b = ",b," , c = ",c," , f[a,b,c] = ",f[a,b,c]]]

Hope this helps.


Ps. Any suggestions to improve the performance of this code will be greatly appreciated. Thank You.



edit:


As per @@Kuba's suggestion, I split the original question in to two. The second part is at Simulated Annealing Parameters and Results



Answer



I found the solution on another website "http://eternaldisturbanceincosmos.wordpress.com/2011/04/27/nminimize-in-mathematica-could-drive-you-insane/" which says "It turns out that NMinimize does not hold its arguments. This means that as the list of arguments is read from left to right, each argument is evaluated and replaced by the result of the evaluation". So I used Hold[] in the NMinimize function that fixed the problem.


Edit:


As suggested by Szabolcs, it worked with ?NumericQ alone and no Hold[]


Comments

Popular posts from this blog

plotting - How to draw lines between specified dots on ListPlot?

I would like to create a plot where I have unconnected dots and some connected. So far, I have figured out how to draw the dots. My code is the following: ListPlot[{{1, 1}, {2, 2}, {3, 3}, {4, 4}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {1, 10}, {2, 11}, {3, 12}, {4,13}, {2.5, 7}}, Ticks -> {{1, 2, 3, 4}, None}, AxesStyle -> Thin, TicksStyle -> Directive[Black, Bold, 12], Mesh -> Full] I have thought using ListLinePlot command, but I don't know how to specify to the command to draw only selected lines between the dots. Do have any suggestions/hints on how to do that? Thank you. Answer One possibility would be to use Epilog with Line : ListPlot[ {{1, 1}, {2, 2}, {3, 3}, {4, 4}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {1, 10}, {2, 11}, {3, 12}, {4, 13}, {2.5, 7}}, Ticks -> {{1, 2, 3, 4}, None}, AxesStyle -> Thin, TicksStyle -> Directive[Black, Bold, 12], Mesh -> Full, Epilog -> { Line[ ...

equation solving - Invert and fit implicitly defined curve

I need to fit an implicitly defined curve. I thought I could get some data out of Solve , and then using FindFit . Therefore, I would like to find the relation the parametric curve defined by $F(x,y)=0$: Solve[-(1/2) + 1/2 (0.41202 BesselK[0, 0.1 Sqrt[x^2 + y^2]] + (0.101483 x BesselK[1, 0.1 Sqrt[x^2 + y^2]])/Sqrt[x^2 + y^2]) == 0, y] But I can't get an output: Solve was unable to solve the system with inexact coefficients or the system obtained by direct rationalization of inexact numbers present in the system. Since many of the methods used by Solve require exact input, providing Solve with an exact version of the system may help. >> Edit: In particular, I would like to fit the data coming from the curve with the expression of another curve, and not with a function $f(x)$. In particular, since this clearly looks like a cardioid , I would like it to fit to something like it. What other strategies could I try?

dynamic - How can I make a clickable ArrayPlot that returns input?

I would like to create a dynamic ArrayPlot so that the rectangles, when clicked, provide the input. Can I use ArrayPlot for this? Or is there something else I should have to use? Answer ArrayPlot is much more than just a simple array like Grid : it represents a ranged 2D dataset, and its visualization can be finetuned by options like DataReversed and DataRange . These features make it quite complicated to reproduce the same layout and order with Grid . Here I offer AnnotatedArrayPlot which comes in handy when your dataset is more than just a flat 2D array. The dynamic interface allows highlighting individual cells and possibly interacting with them. AnnotatedArrayPlot works the same way as ArrayPlot and accepts the same options plus Enabled , HighlightCoordinates , HighlightStyle and HighlightElementFunction . data = {{Missing["HasSomeMoreData"], GrayLevel[ 1], {RGBColor[0, 1, 1], RGBColor[0, 0, 1], GrayLevel[1]}, RGBColor[0, 1, 0]}, {GrayLevel[0], GrayLevel...