Local Optimization With Multiple Starting Points

Ideas
1

Using Version 11’s link to Ipopt, one can write a local minimization function which can take mulitple starting points and bounds on the variables and also return the Lagrange multipliers for the constraints and the variable bounds and the maximum constraint violation, as shown below

In[5]:= imin[
 x^2 - y^2, {Cos[x - y] >= 0.5}, {{x, -5, {.2, 1}, 
   5}, {y, -5, {-4, -1}, 5}}]

Out[5]= {{-24.9443, {x -> 0.235988, y -> -5.}, 
  "conslagmult" -> {0 <= -0.5 + 
       Cos[x - y] <= \[Infinity] -> -0.54499}, 
  "bndlagmult" -> {x -> {4.78912*10^-10, 5.25626*10^-10}, 
    y -> {10.472, 2.50587*10^-10}}, "maxconsviol" -> 4.84126*10^-8, 
  "Solve_Succeeded"}, {-9.37535, {x -> -3.9528, y -> -5.}, 
  "conslagmult" -> {0 <= -0.5 + 
       Cos[x - y] <= \[Infinity] -> -9.12861}, 
  "bndlagmult" -> {x -> {2.39404*10^-9, 2.79887*10^-10}, 
    y -> {2.0944, 2.50574*10^-10}}, "maxconsviol" -> 0, 
  "Solve_Succeeded"}}

Here are the corresponding results from FindMinimum

In[6]:= FindMinimum[{x^2 - y^2, 
  Cos[x - y] >= 0.5, -5 <= x <= 5, -5 <= y <= 5}, {{x, .2}, {y, -4}}]

Out[6]= {-24.9443, {x -> 0.235988, y -> -5.}}

In[7]:= FindMinimum[{x^2 - y^2, 
  Cos[x - y] >= 0.5, -5 <= x <= 5, -5 <= y <= 5}, {{x, 1}, {y, -1}}]

Out[7]= {-9.37535, {x -> -3.9528, y -> -5.}}
1 Like
2

Thanks for sharing! Does FindMinimum also call (internally) Ipopt functionality? or is that completely different?

3

Method → “IPOPT” is an option for FindMinimum. I suspect it is the default for constrained problems.

1 Like