Word for "Approximately Optimize"?

Solution 1:

It depends on context, of course, but if approximate optimization sounds too odd or flowery for your tastes, you could try words like:

• calibrate
• adjust
• attune

When used in conjunction with various equipments, those words imply that performance or measurement is being improved – with the optimum being ideal, though not necessarily attained.

The word tweak can also be used, though that sounds more informal to my ears.

Solution 2:

Satisfice

You'll run into this in economics. The idea is that people don't pull out a laptop with Matlab on it on order to calculate what to do; they merely have a short think in order to come to a solution that satisfies some conditions, and make do with whatever optimizations are easily to hand.