Fix a bug and improve dircon squatting script

[yminchen]

Fix scaling bugs, rename a variable, use bounding box constraint, scale the cost and provide initial guesses for slack/time variables

[yminchen]

What's the effect on benchmark runtime, if any?

Here is what I observed about the solve time:

• Fixing the constraint scaling bug: solve time barely changed
• Using bounding box constraints: slightly decreased
• Scaling the cost: increased by ~50%
• Providing initial guesses for the slack and time variables: decreased by 50~100%

I am not sure why scaling the cost didn't help to improve the solve time in this example

[mposa]

• Scaling the cost: increased by ~50%

So why scale the cost in this PR?

• Providing initial guesses for the slack and time variables: decreased by 50~100%

100% reduction = instantaneous?

[yminchen]

Ah I made some mistakes in the previous comment. I benchmarked the solve time more carefully this time and here is the result:

• Fixing the constraint scaling bug: solve time increased by ~40 seconds
• Using bounding box constraints: increased by ~10 seconds
• Providing initial guesses for the slack and time variables: decreased by ~10 seconds
• Scaling the cost: decreased by ~30 seconds

The above changes are listed in order. With a different order, we might get a different solve time result. For example, if we only do the latter 3 bullet points, the solve time is ~1600 seconds...

Interestingly, the unintentional scaling "bug" improved the solving speed although it worsened the optimal cost by about 1%. I call it a bug here because it caused the condition number of the constraint gradient to be bad. Should we still merge this PR?

[mposa]

1/500 is a pretty big scale factor to include if it doesn't actually improve performance. Might be worth dialing it back to see what happens.

[yminchen]

Ok, I'll close this PR for now