I have struggled for a while (years?) with getting some form of intuition for duality results in linear programming. Today I finally found this gem:

http://cstheory.stackexchange.com/q/16206/20762.

In case you want to form some intuition about duality, I couldn’t encourage you enough to carefully read through Neal Young’s comment and Mike Spivey’s answer to the above question on cstheory.stackexchange.