Plotting FindMinimum internal steps

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MODERATORS’ NOTE: full code and function definition are available at the end of the post.

For problems in 2 or 3 dimensions, step monitor can be used to plot the variable values of the internal steps taken by FindMinimum. A different approach, which is not limited to 3 dimensions, is to plot the value of the objective function vs the maximum constraint violation. A function FindMinimumWithPlot is defined in the attached notebook and some simple examples are shown here.

FindMinimumWithPlot[{x + y, x^2 + y^2 == 1}, {{x, 1}, {y, 2}}]

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{-1.41421, {x -> -0.707107, y -> -0.707107}, 
 "max constraint violation" -> 2.3684*10^-9}

FindMinimumWithPlot[{x^2 - y^2, 
  Cos[x - y] >= 0.5, -5 <= x <= 5, -5 <= y <= 5}, {{x, 4}, {y, -4}}]

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{-24.9443, {x -> 0.235988, y -> -5.}, 
 "max constraint violation" -> 4.13187*10^-8}

FindMinimumWithPlot[{x + y + z, (x - 1/2)^2 + y^2 + z^2 == 
   1, (x + 1/2)^2 + y^2 + z^2 == 1}, {{x, 1}, {y, -1}, {z, 1}}, 
 Method -> "IPOPT"]

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{-1.22474, {x -> 1.91952*10^-17, y -> -0.612372, z -> -0.612372}, 
 "max constraint violation" -> 1.42258*10^-12}
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Plotting NMinimize internal steps
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The procedure consists of using compiled functions (for speed) to have StepMonitor return the maximum constraint violation and objective function value at each step. These are then partitioned into pairs and plotted using the Arrow function.