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Black box optimization
Posted:
Apr 28, 2012 5:32 AM


Is there any black box optimization of user defined nonpolynomial functions in Mathematica? I.e.
I want to minimize fm[x] between 0.010 and 0.060. The goal is to fit the data with mx+b. This requires two points. The first point in the data has to be zero or first element shown below. The other single point must allow a fit with minimum error between the original data points and the new data points generated from an mx+b approximation.
fm[mPt_]:=Module[{mMinFit,mFit,mError,x,InData}, InData={{0.`,0.3457378`},{0.005005030108147661`,0.5947282`},{0.010167934319260488`,1.110245`},{0.015746789471210974`,1.753068`},{0.019877754878728275`,2.26061`},{0.025058168807019193`,2.891833`},{0.029851036834650214`,3.470055`},{0.03486106617079409`,4.088596`},{0.04009652061250109`,4.721034`},{0.04501992441075972`,5.31037`},{0.049993105670535644`,5.912859`},{0.054948450286312706`,6.513352`},{0.06007028590992394`,7.144364`}}; (* Use mMinFit to select Y value for selected point *) mMinFit=Fit[Select[InData, #[[1]] > 0.01&],{1,x},x]; (* Generate fit between new fit between first point and new test point *) mFit=Fit[{First@InData,{mPt,mMinFit /. x>mPt}},{1,x},x]; (* subtract real data from points generated by new curve *) mError=Total@Table[Abs@(m[[2]]mFit /. x >m[[1]]),{m,InData}] ]
Calling fm[0.01] calculates the fit using {{0.`,0.3457378`},{0.01,InterpValue} as the two points mx+b must pass through. It then returns the Abs[] of the difference between the original points (InData) and the interpolated points based on original x values. This is intended to be the error function. Minimizing fm[x] should give the best possible choice of x to calibrate with.
I can always fall back to:
m=Table[{i,fm[i]},{i,0.010,0.060,0.00001}]; First@Sort[m,#1[[2]] < #2[[2]]&]
Out:= {0.04474,2.13522}
Here is a decent graph of the issue:
ListPlot[Table[fm[i], {i, 0.010, 0.060, 0.001}], Joined > True]
I thought I found a better way in Mathematica before...
Paul McHale  Electrical Engineer, Energetics Systems  Excelitas Technologies Corp.
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