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Minimize over Arbitrary Regions
Mathematica 6 adds major new algorithms for minimizing nonlinear functions over arbitrary regions.
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Table[With[{f = Sin[x + Cos[y]], 

   cons = x^2 < y^3 + 5 && y^2 < x^3 + u}, 

  Show[ContourPlot[f, {x, -2, 3}, {y, -2, 3}, 

    RegionFunction -> Function @@ {{x, y}, cons}, 

    ColorFunction -> "ThermometerColors"], 

   Graphics[{Blue, 

     Disk[{x, y} /. 

       Last[FindMinimum[{f, cons}, {x, y}]], .2]}]]], {u, -1, 2}]
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