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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[]


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