Constrained Optimization with Interpolating Functions

Question

My goal is to use a gradient descent type method to maximize interpolating function1 with respect to the constraint that interpolating function2 <= 0.5. I am working with 4D data (please see below).

I created the two interpolating functions as follows:

f1 = Interpolation[data1, InterpolationOrder -> 1]
f2 = Interpolation[data2, InterpolationOrder -> 1]

And then I used the code outlined in this resource https://library.wolfram.com/infocenter/Books/8506/ConstrainedOptimization.pdf:

NMinimize[{-f2, f1 <=  0.5}, {x, y, z}]
Minimize[{-f2, f1 <=  0.5}, {x, y, z}]
FindMinimum[{-f2, f1 <=  0.5}, {x, y, z}]

NMinimize gives the error message: "NMinimize::bcons: The following constraints are not valid: {InterpolatingFunction[{{4.,8.},{0.5,7.},{0.7,2.}},{5,4225,0,{954,0,6},{2,2,2},0,0,0,0,Automatic,{},{},False},{<<23>>[{<<1>>},{<<1>>},{<<1>>}]},{0.750342,0.64151,0.622998,0.730189,0.236956,0.0920386,0.0884832,<<37>>,0.000752695,0.000144944,0.0000162,3.7210^-6,2.6710^-6,0.316358,<<904>>},{Automatic}]<=<<4>>}. Constraints should be equalities, inequalities, or domain specifications involving the variables."

Minimize gives no resulting value and just outputs the same thing that I input.

Find Minimum gives the error "FindInstance::exvar: The system contains a nonconstant expression InterpolatingFunction[{{4.,8.},{0.5,7.},{0.7,2.}},{5,4225,0,{954,0,6},{2,2,2},0,0,0,0,Automatic,{},{},False},{<<1>>},{0.750342,0.64151,0.622998,0.730189,0.236956,0.0920386,0.0884832,0.948879,<<35>>,0.00922568,0.000752695,0.000144944,0.0000162,3.7210^-6,2.6710^-6,0.316358,<<904>>},{Automatic}] independent of variables {[FormalX],[FormalY],[FormalZ]}."

Any help figuring out how to do this type of optimization will be very greatly appreciated. I have never done this in Mathematica and I am very confused.

Here is some of the data that I used in case anyone wants to test out what I'm talking about:

data1 = {{{4.0, 2.5, 0.7}, 0.750341814}, {{4.0, 2.75, 0.7}, 
   0.641510140}, {{4.0, 3.0, 0.7}, 0.622998451}, {{4.5, 2.3, 0.7}, 
   0.730188891}, {{4.5, 2.6, 0.7}, 0.236956251}, {{4.5, 3.2, 0.7}, 
   0.092038571}, {{4.5, 3.5, 0.7}, 0.088483203}, {{5.0, 2.25, 0.7}, 
   0.948879277}, {{5.0, 2.6, 0.7}, 0.190624075}, {{5.0, 2.95, 0.7}, 
   0.135275036}, {{5.0, 3.3, 0.7}, 0.032763743}, {{5.0, 3.65, 0.7}, 
   0.029430211}, {{5.0, 4.0, 0.7}, 0.027537075}, {{5.5, 2.5, 0.7}, 
   0.281232137}, {{5.5, 2.9, 0.7}, 0.054980707}, {{5.5, 3.3, 0.7}, 
   0.023085488}, {{5.5, 3.7, 0.7}, 0.031317836}, {{5.5, 4.1, 0.7}, 
   0.010820878}, {{5.5, 4.5, 0.7}, 0.010161860}, {{6.0, 2.3, 0.7}, 
   0.397089577}, {{6.0, 2.75, 0.7}, 0.074292346}, {{6.0, 3.2, 0.7}, 
   0.015918433}, {{6.0, 3.65, 0.7}, 0.004494633}, {{6.0, 4.1, 0.7}, 
   0.002195262}, {{6.0, 4.55, 0.7}, 0.001750180}, {{6.0, 5.0, 0.7}, 
   0.001701018}, {{6.5, 2.5, 0.7}, 0.322173220}, {{6.5, 3.0, 0.7}, 
   0.037894300}, {{6.5, 3.5, 0.7}, 0.005178418}, {{6.5, 4.0, 0.7}, 
   0.001170227}, {{6.5, 4.5, 0.7}, 0.001044620}, {{6.5, 5.0, 0.7}, 
   0.000228311}, {{6.5, 5.5, 0.7}, 0.000204873}, {{7.0, 2.15, 0.7}, 
   0.799562745}, {{7.0, 2.7, 0.7}, 0.169379167}, {{7.0, 3.25, 0.7}, 
   0.019800399}, {{7.0, 3.8, 0.7}, 0.001902681}, {{7.0, 4.35, 0.7}, 
   0.000501842}, {{7.0, 4.9, 0.7}, 0.000126206}, {{7.0, 5.45, 0.7}, 
   0.000026700}, {{7.0, 6.0, 0.7}, 0.000020600}, {{7.5, 2.3, 0.7}, 
   1.215594070}, {{7.5, 2.9, 0.7}, 0.050740772}, {{7.5, 3.5, 0.7}, 
   0.009225684}, {{7.5, 4.1, 0.7}, 0.000752695}, {{7.5, 4.7, 0.7}, 
   0.000144944}, {{7.5, 5.3, 0.7}, 0.000016200}, {{7.5, 5.9, 0.7}, 
   0.000003720}, {{7.5, 6.5, 0.7}, 0.000002670}, {{8.0, 2.45, 0.7}, 
   0.316357500}, {{8.0, 3.1, 0.7}, 0.049589355}, {{8.0, 3.75, 0.7}, 
   0.001624453}, {{8.0, 4.4, 0.7}, 0.000321852}, {{8.0, 5.05, 0.7}, 
   0.000043800}, {{8.0, 5.7, 0.7}, 0.000003320}, {{8.0, 6.35, 0.7}, 
   0.000001040}, {{8.0, 7.0, 0.7}, 0.000000425}, {{4.0, 2.0, 0.83}, 
   1.123030977}, {{4.0, 2.25, 0.83}, 0.643862594}, {{4.0, 2.5, 0.83}, 
   0.494527641}, {{4.0, 2.75, 0.83}, 0.460772203}, {{4.0, 3.0, 0.83}, 
   0.449952666}, {{4.5, 2.0, 0.83}, 1.177597462}, {{4.5, 2.3, 0.83}, 
   0.246286304}, {{4.5, 2.6, 0.83}, 0.126303219}, {{4.5, 2.9, 0.83}, 
   0.805843164}, {{4.5, 3.2, 0.83}, 0.064124542}, {{4.5, 3.5, 0.83}, 
   0.063221002}, {{5.0, 1.9, 0.83}, 0.959095797}, {{5.0, 2.25, 0.83}, 
   0.395763200}, {{5.0, 2.6, 0.83}, 0.068053440}, {{5.0, 2.95, 0.83}, 
   0.058924421}, {{5.0, 3.3, 0.83}, 0.018638068}, {{5.0, 3.65, 0.83}, 
   0.016715326}, {{5.0, 4.0, 0.83}, 0.016485697}, {{5.5, 2.1, 0.83}, 
   0.801945073}, {{5.5, 2.5, 0.83}, 0.121609842}, {{5.5, 2.9, 0.83}, 
   0.022526237}, {{5.5, 3.3, 0.83}, 0.010862152}, {{5.5, 3.7, 0.83}, 
   0.010286207}, {{5.5, 4.1, 0.83}, 0.007491423}, {{5.5, 4.5, 0.83}, 
   0.007211895}, {{6.0, 2.3, 0.83}, 0.160576165}, {{6.0, 2.75, 0.83}, 
   0.029644050}, {{6.0, 3.2, 0.83}, 0.007040139}, {{6.0, 3.65, 0.83}, 
   0.003064983}, {{6.0, 4.1, 0.83}, 0.002347885}, {{6.0, 4.55, 0.83}, 
   0.002231154}, {{6.0, 5.0, 0.83}, 0.002229741}, {{6.5, 2.0, 0.83}, 
   0.743003719}, {{6.5, 2.5, 0.83}, 0.091745464}, {{6.5, 3.0, 0.83}, 
   0.011634382}, {{6.5, 3.5, 0.83}, 0.001858806}, {{6.5, 4.0, 0.83}, 
   0.000625568}, {{6.5, 4.5, 0.83}, 0.000476692}, {{6.5, 5.0, 0.83}, 
   0.000270385}, {{6.5, 5.5, 0.83}, 0.000263443}, {{7.0, 2.15, 0.83}, 
   0.289692249}, {{7.0, 2.7, 0.83}, 0.070986865}, {{7.0, 3.25, 0.83}, 
   0.005445419}, {{7.0, 3.8, 0.83}, 0.000569259}, {{7.0, 4.35, 0.83}, 
   0.000184639}, {{7.0, 4.9, 0.83}, 0.000050600}, {{7.0, 5.45, 0.83}, 
   0.000019100}, {{7.0, 6.0, 0.83}, 0.000017700}, {{7.5, 2.3, 0.83}, 
   0.427847385}, {{7.5, 2.9, 0.83}, 0.016100380}, {{7.5, 3.5, 0.83}, 
   0.003665870}, {{7.5, 4.1, 0.83}, 0.000201233}, {{7.5, 4.7, 0.83}, 
   0.000060600}, {{7.5, 5.3, 0.83}, 0.000005190}, {{7.5, 5.9, 0.83}, 
   0.000002460}, {{7.5, 6.5, 0.83}, 0.000002090}, {{8.0, 2.45, 0.83}, 
   0.096275542}, {{8.0, 3.1, 0.83}, 0.017935820}, {{8.0, 3.75, 0.83}, 
   0.000604182}, {{8.0, 4.4, 0.83}, 0.000079200}, {{8.0, 5.05, 0.83}, 
   0.000017500}, {{8.0, 5.7, 0.83}, 0.000000999}, {{8.0, 6.35, 0.83}, 
   0.000000476}, {{8.0, 7.0, 0.83}, 0.000000303}, {{4.0, 2.25, 0.96}, 
   1.177242345}, {{4.0, 2.5, 0.96}, 0.903436502}, {{4.0, 2.75, 0.96}, 
   0.332421564}, {{4.0, 3.0, 0.96}, 0.328294859}, {{4.5, 2.0, 0.96}, 
   0.610951015}, {{4.5, 2.3, 0.96}, 0.673977792}, {{4.5, 2.6, 0.96}, 
   0.108851985}, {{4.5, 2.9, 0.96}, 0.125798036}, {{4.5, 3.2, 0.96}, 
   0.164368921}, {{4.5, 3.5, 0.96}, 0.083522130}, {{5.0, 2.25, 0.96}, 
   0.173705123}, {{5.0, 2.6, 0.96}, 0.414522040}, {{5.0, 2.95, 0.96}, 
   0.197962721}, {{5.0, 3.3, 0.96}, 0.025581569}, {{5.0, 3.65, 0.96}, 
   0.026675785}, {{5.0, 4.0, 0.96}, 0.025212890}, {{5.5, 2.1, 0.96}, 
   0.441425969}, {{5.5, 2.5, 0.96}, 0.044996441}, {{5.5, 2.9, 0.96}, 
   0.306236219}, {{5.5, 3.3, 0.96}, 0.010891276}, {{5.5, 3.7, 0.96}, 
   0.006561283}, {{5.5, 4.1, 0.96}, 0.000851567}, {{5.5, 4.5, 0.96}, 
   0.000718190}, {{6.0, 1.85, 0.96}, 0.807149508}, {{6.0, 2.3, 0.96}, 
   0.065261072}, {{6.0, 2.75, 0.96}, 0.062784999}, {{6.0, 3.2, 0.96}, 
   0.073790906}, {{6.0, 3.65, 0.96}, 0.013393139}, {{6.0, 4.1, 0.96}, 
   0.002261782}, {{6.0, 4.55, 0.96}, 0.001243003}, {{6.0, 5.0, 0.96}, 
   0.001121045}, {{6.5, 2.0, 0.96}, 1.145752994}, {{6.5, 2.5, 0.96}, 
   0.190715368}, {{6.5, 3.0, 0.96}, 0.039221201}, {{6.5, 3.5, 0.96}, 
   0.023321520}, {{6.5, 4.0, 0.96}, 0.000322579}, {{6.5, 4.5, 0.96}, 
   0.000220124}, {{6.5, 5.0, 0.96}, 0.000187539}, {{6.5, 5.5, 0.96}, 
   0.000196152}, {{7.0, 2.15, 0.96}, 0.554272538}, {{7.0, 2.7, 0.96}, 
   0.028907939}, {{7.0, 3.25, 0.96}, 0.009721355}, {{7.0, 3.8, 0.96}, 
   0.002695021}, {{7.0, 4.35, 0.96}, 0.000084600}, {{7.0, 4.9, 0.96}, 
   0.000142253}, {{7.0, 5.45, 0.96}, 0.000037600}, {{7.0, 6.0, 0.96}, 
   0.000034100}};



   data2 = {{{4.0, 2.5, 0.7}, 1825638.3969999999}, {{4.0, 2.75, 0.7}, 
   1724222.467}, {{4.0, 3.0, 0.7}, 1642032.871}, {{4.5, 2.3, 0.7}, 
   1854987.9919999999}, {{4.5, 2.6, 0.7}, 
   1705997.1269999999}, {{4.5, 3.2, 0.7}, 
   1492211.864}, {{4.5, 3.5, 0.7}, 1416414.175}, {{5.0, 2.25, 0.7}, 
   1838521.2130000002}, {{5.0, 2.6, 0.7}, 
   1657210.419}, {{5.0, 2.95, 0.7}, 1515504.514}, {{5.0, 3.3, 0.7}, 
   1403475.8590000002}, {{5.0, 3.65, 0.7}, 
   1313632.943}, {{5.0, 4.0, 0.7}, 1243469.29}, {{5.5, 2.5, 0.7}, 
   1673804.5769999998}, {{5.5, 2.9, 0.7}, 
   1499202.05}, {{5.5, 3.3, 0.7}, 1363569.187}, {{5.5, 3.7, 0.7}, 
   1256514.6809999999}, {{5.5, 4.1, 0.7}, 
   1171796.743}, {{5.5, 4.5, 0.7}, 1105440.165}, {{6.0, 2.3, 0.7}, 
   1761443.8190000001}, {{6.0, 2.75, 0.7}, 
   1537810.2040000001}, {{6.0, 3.2, 0.7}, 
   1370202.236}, {{6.0, 3.65, 0.7}, 1240452.455}, {{6.0, 4.1, 0.7}, 
   1137210.962}, {{6.0, 4.55, 0.7}, 1056733.831}, {{6.0, 5.0, 0.7}, 
   994385.1308}, {{6.5, 2.5, 0.7}, 1641853.405}, {{6.5, 3.0, 0.7}, 
   1425165.07}, {{6.5, 3.5, 0.7}, 1262873.913}, {{6.5, 4.0, 0.7}, 
   1137578.723}, {{6.5, 4.5, 0.7}, 1037990.926}, {{6.5, 5.0, 0.7}, 
   961077.5862}, {{6.5, 5.5, 0.7}, 902696.5156}, {{7.0, 2.15, 0.7}, 
   1836694.5659999999}, {{7.0, 2.7, 0.7}, 
   1537852.989}, {{7.0, 3.25, 0.7}, 
   1328450.5559999999}, {{7.0, 3.8, 0.7}, 
   1171256.773}, {{7.0, 4.35, 0.7}, 1050080.866}, {{7.0, 4.9, 0.7}, 
   954035.877}, {{7.0, 5.45, 0.7}, 880665.0223}, {{7.0, 6.0, 0.7}, 
   825700.3843}, {{7.5, 2.3, 0.7}, 1739780.707}, {{7.5, 2.9, 0.7}, 
   1447164.261}, {{7.5, 3.5, 0.7}, 1241853.154}, {{7.5, 4.1, 0.7}, 
   1089986.5559999999}, {{7.5, 4.7, 0.7}, 
   973268.6319}, {{7.5, 5.3, 0.7}, 882716.8063}, {{7.5, 5.9, 0.7}, 
   812719.8843}, {{7.5, 6.5, 0.7}, 759854.6725}, {{8.0, 2.45, 0.7}, 
   1647668.73}, {{8.0, 3.1, 0.7}, 1366059.112}, {{8.0, 3.75, 0.7}, 
   1168252.432}, {{8.0, 4.4, 0.7}, 1020992.728}, {{8.0, 5.05, 0.7}, 
   908604.2184}, {{8.0, 5.7, 0.7}, 820844.7401}, {{8.0, 6.35, 0.7}, 
   753595.9017}, {{8.0, 7.0, 0.7}, 703633.5985}, {{4.0, 2.0, 0.83}, 
   2018289.761}, {{4.0, 2.25, 0.83}, 1890663.007}, {{4.0, 2.5, 0.83}, 
   1788071.095}, {{4.0, 2.75, 0.83}, 1701044.687}, {{4.0, 3.0, 0.83}, 
   1632910.4419999998}, {{4.5, 2.0, 0.83}, 
   1934243.685}, {{4.5, 2.3, 0.83}, 1779632.375}, {{4.5, 2.6, 0.83}, 
   1657469.8269999998}, {{4.5, 2.9, 0.83}, 
   1557370.45}, {{4.5, 3.2, 0.83}, 1476857.989}, {{4.5, 3.5, 0.83}, 
   1410963.305}, {{5.0, 1.9, 0.83}, 1940867.365}, {{5.0, 2.25, 0.83}, 
   1746026.48}, {{5.0, 2.6, 0.83}, 1596546.918}, {{5.0, 2.95, 0.83}, 
   1477049.404}, {{5.0, 3.3, 0.83}, 1381061.372}, {{5.0, 3.65, 0.83}, 
   1301990.0729999999}, {{5.0, 4.0, 0.83}, 
   1241398.777}, {{5.5, 2.1, 0.83}, 1785463.573}, {{5.5, 2.5, 0.83}, 
   1594840.5790000001}, {{5.5, 2.9, 0.83}, 
   1446892.08}, {{5.5, 3.3, 0.83}, 
   1331446.3059999999}, {{5.5, 3.7, 0.83}, 
   1238565.025}, {{5.5, 4.1, 0.83}, 
   1163619.8090000001}, {{5.5, 4.5, 0.83}, 
   1105297.283}, {{6.0, 2.3, 0.83}, 
   1655369.4740000002}, {{6.0, 2.75, 0.83}, 
   1469443.386}, {{6.0, 3.2, 0.83}, 1325426.857}, {{6.0, 3.65, 0.83}, 
   1212210.146}, {{6.0, 4.1, 0.83}, 1122233.172}, {{6.0, 4.55, 0.83}, 
   1050716.784}, {{6.0, 5.0, 0.83}, 995461.9561}, {{6.5, 2.0, 0.83}, 
   1798291.485}, {{6.5, 2.5, 0.83}, 1546739.957}, {{6.5, 3.0, 0.83}, 
   1363093.63}, {{6.5, 3.5, 0.83}, 1222583.733}, {{6.5, 4.0, 0.83}, 
   1111645.925}, {{6.5, 4.5, 0.83}, 1024967.692}, {{6.5, 5.0, 0.83}, 
   956236.3007}, {{6.5, 5.5, 0.83}, 903703.0732}, {{7.0, 2.15, 0.83}, 
   1700586.6619999998}, {{7.0, 2.7, 0.83}, 
   1453964.937}, {{7.0, 3.25, 0.83}, 
   1273237.4270000001}, {{7.0, 3.8, 0.83}, 
   1135337.705}, {{7.0, 4.35, 0.83}, 1027855.275}, {{7.0, 4.9, 0.83}, 
   943468.5092}, {{7.0, 5.45, 0.83}, 877501.7975}, {{7.0, 6.0, 0.83}, 
   827340.708}, {{7.5, 2.3, 0.83}, 1614584.554}, {{7.5, 2.9, 0.83}, 
   1371301.265}, {{7.5, 3.5, 0.83}, 1194308.89}, {{7.5, 4.1, 0.83}, 
   1059602.106}, {{7.5, 4.7, 0.83}, 955129.4112}, {{7.5, 5.3, 0.83}, 
   873440.8529}, {{7.5, 5.9, 0.83}, 
   810232.4509999999}, {{7.5, 6.5, 0.83}, 
   762267.3347}, {{8.0, 2.45, 0.83}, 1537762.459}, {{8.0, 3.1, 0.83}, 
   1297367.76}, {{8.0, 3.75, 0.83}, 1123677.977}, {{8.0, 4.4, 0.83}, 
   992427.6695}, {{8.0, 5.05, 0.83}, 890887.4116}, {{8.0, 5.7, 0.83}, 
   811914.6927}, {{8.0, 6.35, 0.83}, 751053.9011}, {{8.0, 7.0, 0.83}, 
   705433.4887}, {{4.0, 2.25, 0.96}, 
   1828024.7130000002}, {{4.0, 2.5, 0.96}, 
   1744444.3830000001}, {{4.0, 2.75, 0.96}, 
   1674720.746}, {{4.0, 3.0, 0.96}, 1616854.769}, {{4.5, 2.0, 0.96}, 
   1829325.034}, {{4.5, 2.3, 0.96}, 1710008.313}, {{4.5, 2.6, 0.96}, 
   1610703.7030000002}, {{4.5, 2.9, 0.96}, 
   1527864.519}, {{4.5, 3.2, 0.96}, 
   1459332.9419999998}, {{4.5, 3.5, 0.96}, 
   1403317.9719999998}, {{5.0, 2.25, 0.96}, 
   1661860.34}, {{5.0, 2.6, 0.96}, 1539826.333}, {{5.0, 2.95, 0.96}, 
   1440650.7680000002}, {{5.0, 3.3, 0.96}, 
   1358658.493}, {{5.0, 3.65, 0.96}, 1291758.576}, {{5.0, 4.0, 0.96}, 
   1237451.801}, {{5.5, 2.1, 0.96}, 1675487.35}, {{5.5, 2.5, 0.96}, 
   1523401.9230000002}, {{5.5, 2.9, 0.96}, 
   1400769.7219999998}, {{5.5, 3.3, 0.96}, 
   1301829.295}, {{5.5, 3.7, 0.96}, 1220936.543}, {{5.5, 4.1, 0.96}, 
   1155585.378}, {{5.5, 4.5, 0.96}, 
   1103793.0729999999}, {{6.0, 1.85, 0.96}, 
   1757955.146}, {{6.0, 2.3, 0.96}, 1561884.72}, {{6.0, 2.75, 0.96}, 
   1407948.372}, {{6.0, 3.2, 0.96}, 1285886.175}, {{6.0, 3.65, 0.96}, 
   1187551.9270000001}, {{6.0, 4.1, 0.96}, 
   1108199.767}, {{6.0, 4.55, 0.96}, 
   1044915.8470000001}, {{6.0, 5.0, 0.96}, 
   995178.0803}, {{6.5, 2.0, 0.96}, 
   1663896.2219999998}, {{6.5, 2.5, 0.96}, 
   1464140.99}, {{6.5, 3.0, 0.96}, 1310108.155}, {{6.5, 3.5, 0.96}, 
   1188448.767}, {{6.5, 4.0, 0.96}, 1091521.031}, {{6.5, 4.5, 0.96}, 
   1013705.87}, {{6.5, 5.0, 0.96}, 952292.5175}, {{6.5, 5.5, 0.96}, 
   904596.5571}, {{7.0, 2.15, 0.96}, 1581377.844}, {{7.0, 2.7, 0.96}, 
   1379491.4209999999}, {{7.0, 3.25, 0.96}, 
   1225531.168}, {{7.0, 3.8, 0.96}, 1105133.229}, {{7.0, 4.35, 0.96}, 
   1009706.7059999999}, {{7.0, 4.9, 0.96}, 
   933709.8657}, {{7.0, 5.45, 0.96}, 873895.2218}, {{7.0, 6.0, 0.96}, 
   828561.4393}};

Answer 1

You could try somthing like this. Using a MeshRegion as domain guarantess that the optimization algorithm won't leave the domain of definition of f1 and f2. (Using f1 and f2 for extrapolation might do crazy things.) Not having the data, I cannot check whether this works...

R1 = ConvexHullMesh[data1[[All, 1]]];
R2 = ConvexHullMesh[data2[[All, 1]]];
R = RegionIntersection[R1, R2];
f1 = Interpolation[data1, InterpolationOrder -> 1];
f2 = Interpolation[data2, InterpolationOrder -> 1];

sol = NMinimize[
        {-f2[x, y, z], f1[x, y, z] <= 0.5}, 
        {x, y, z} \[Element] R
      ]

X = {x, y, z} /. sol[[2]];
f1 @@ X <= 0.5
RegionMember[R, X]

True

True

For some reason, the kernel crashes on this. But as piecwise-linear functions on some tetrahedral mesh are minimized: Those attain there extrema also on the set of vertices -- and the latter are precisely the points in the datasets. So this should also find the minimum:

pos = Position[data1[[All, 2]], _?(# <= 0.5 &)];
minind = Ordering[Extract[-data2[[All, 2]], pos], -1, 1][[1]];
minimum = data2[[pos[[minind, 1]], 1]]
minvalue = data2[[pos[[minind, 1]], 2]]

{8., 7., 2.}

722571.