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Chaos in constrained optimization

Animations of two circle packings are shown.  The circles have radii of Range[10] raised to -1/2 power (1.0, 0.707, ....).
The objective function is minimization of the circumscribing circle.  The only difference between the two is that circle 4,
in the animation on the right, starts 0.01 to the right of its location in the animation on the left. As a result, it ends up in
a different place in the final result.
POSTED BY: Frank Kampas
2 Replies
The calculations are done with FindMinimum, using EvaluationMonitor to get the coordinates of the circle centers at every step.  The objective function is radius of the circumscribing circle, r0.  { x, y } is the center of circle i.  A simplified version of the Mathematica code is
 n = 10;
 
 Do[r[i] = i^(-1/2), {i, n}];
 
 vars = Join[{r0}, Flatten[Table[{x[i], y[i]}, {i, n}]]];
 
 minimaxcons = Thread[r0 >= Table[r[i] + Sqrt[x[i]^2 + y[i]^2], {i, n}]];
 
 overlapcons = Flatten[Table[(r[i] + r[j] )^2 <=  (x[i] - x[j])^2 + (y[i] - y[j])^2, {i, n - 1}, {j, i + 1, n}]];

FindMinimum[{r0, Sequence @@ Join[minimaxcons, overlapcons]}, Thread[{vars, start}]]
A "minimax" approach is used.  start is the the starting values of the variables, which I chose to illustrate how a small difference could cause a much bigger difference in the result.

I think this illustrates the difficulties of doing global optimization in a problem with a large number of local minimum.  I believe it occurs due to the line search nature of the constrained optimization.  However, any other method would be much slower.
POSTED BY: Frank Kampas
Frank, is it possible to share code for these animations? Also, are you noticing this as an interesting phenomenon or reporting undesired behavior?
POSTED BY: Sam Carrettie
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