I always forget what does pareto optimality mean. I reviewed the slides of convex optimization course by Stephen Boyd recently and found the following very intuitive example. The curve below shows that to maintain a certain production volume, a factory needs to allocate its resources (fuel and labor) in the region 
. Essentially, we can trade off fuel and labor and yet to maintain the production volume. Of course, any point in the interior of is not optimal since we can lower fuel (by moving down) or lower labor (by moving to the left). But even at the boundary, those points may not be optimal. In the curve below, points 
, 
, and 
are optimal but 
and 
are not. Because for the latter two, apparently we can reduce labor cost further by simply moving to .
Interestingly, I have encountered these sorts of things many times before in rate-distortion and capacity-power functions. But I didn’t know the fancy jargoon “pareto” optimal at that time. And just as in the above communication problems, apparently we can construct the pareto optimal curve as the convex hull of the boundary. And any point that does not lie on the original boundary can be reached by time-sharing.
