Wolfram Technology Guide: Constrained Nonlinear Optimization  previous | next 
Handle Systems of Nonlinear Constraints
Mathematica can minimize nonlinear functions with many variables and large numbers of nonlinear constraints. Here it finds the ellipse of minimum area that encloses a given collection of points.
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cloud = (# {1, .5}) & /@ RandomReal[NormalDistribution[], {200, 2}];
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sol = FindMinimum[{r^4/(t^2 s^2), 

   Map[t^2 (#[[1]] - a)^2 + s^2 (#[[2]] - b)^2 < r^2 &, cloud]}, {r, 

   a, b, t, s}]
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RegionPlot[(t^2 (x - a)^2 + s^2 (y - b)^2 < r^2) /. Last[sol], {x, -3,

   3}, {y, -3, 3}, Epilog -> Point[cloud]]
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