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Black box optimization

  • To: mathgroup at smc.vnet.net
  • Subject: [mg126282] Black box optimization
  • From: "McHale, Paul" <Paul.McHale at excelitas.com>
  • Date: Sat, 28 Apr 2012 05:29:17 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com

Is there any black box optimization of user defined non-polynomial 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.

Phone:   +1 937.865.3004   |   Fax:  +1 937.865.5170   |   Mobile:   +1 937.371.2828
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Paul.McHale at Excelitas.com
www.excelitas.com



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