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FindMinimum Variable Independence Puzzle

Using an example from the NMinimize documentation:

f[x, y] = E^Sin[50 x] + Sin[60 E^y] + Sin[70 Sin[x]] + Sin[Sin[80 y]] - Sin[10 (x + y)] + 1/4 (x^2 + y^2);

I use FindMinimum from two different starting points:

In[7]:= FindMinimum[f[x, y], {{x, 1}, {y, 1}}]

Out[7]= {-0.342853, {x -> 0.953282, y -> 0.291998}}

In[8]:= FindMinimum[f[x, y], {{x, -1}, {y, -1}}]

Out[8]= {-1.40759, {x -> -1.66754, y -> 0.67725}}

The I try to do both starting points at once:

In[9]:= FindMinimum[ f[x1, y1] + f[x2, y2], {{x1, 1}, {y1, 1}, {x2, -1}, {y2, -1}}]

Out[9]= {-0.674649, {x1 -> 0.994175, y1 -> 0.993418, x2 -> -1.00898, y2 -> -0.957112}}

and get a different answer. It seems to me the minimization of f[x1,y1] should be independent of the minimization of f[x2,y2]
POSTED BY: Frank Kampas
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On reflection, I think what's coupling the different variables together is the backtracking in the line search. I tried a similar experiment with the COIN-OR solver IPOPT

https://projects.coin-or.org/Ipopt

and got the independence I was expecting. 

callIpOpt[f[x, y], {}, {{x, -2, 1, 2}, {y, -2, 1, 2}}]

{-0.618508, {x -> 0.994175, y -> 0.993418}, "cons values" -> {}, 
 "var lb \[Lambda]s" -> {1.24674*10^-9, 1.24706*10^-9}, 
 "var ub \[Lambda]s" -> {3.71136*10^-9, 3.70856*10^-9}, 
 "cons \[Lambda]s" -> {}, "Solve_Succeeded"}

callIpOpt[f[x, y], {}, {{x, -2, -1, 2}, {y, -2, -1, 2}}]

{-0.0561414, {x -> -1.00898, y -> -0.957112}, "cons values" -> {}, 
 "var lb \[Lambda]s" -> {2.5286*10^-9, 2.40285*10^-9}, 
 "var ub \[Lambda]s" -> {8.32809*10^-10, 8.47416*10^-10}, 
 "cons \[Lambda]s" -> {}, "Solve_Succeeded"}

callIpOpt[
 f[x1, y1] + 
  f[x2, y2], {}, {{x1, -2, 1, 2}, {y1, -2, 1, 2}, {x2, -2, -1, 
   2}, {y2, -2, -1, 2}}]

{-0.674649, {x1 -> 0.994175, y1 -> 0.993418, x2 -> -1.00898, 
  y2 -> -0.957112}, "cons values" -> {}, 
 "var lb \[Lambda]s" -> {1.24674*10^-9, 1.24706*10^-9, 3.76679*10^-9, 
   3.57946*10^-9}, 
 "var ub \[Lambda]s" -> {3.71136*10^-9, 3.70856*10^-9, 1.24061*10^-9, 
   1.26237*10^-9}, "cons \[Lambda]s" -> {}, "Solve_Succeeded"}

Thanks for suggesting the formatting tool. I hadn't realized cutting and pasting dropped things, like underscores.
POSTED BY: Frank Kampas
POSTED BY: David Reiss
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