I think bounded is a very unfortunate word. It doesn’t really capture the inherent differences in solvability. Not surprising though, without actual computer implementations it was hard to grasp what could be solved and how fast. In the absence of that it is hard to see that linear/quadratic/conic/convex/etc. is the important feature. (Just a speculation, but I think Morse was referring to the fact that the optimal solution is obtained as the intersection of some of the constraints vs inside the feasible set, where it could be located with calculus.)
I guess these days it would be called constrained optimization, which is an equally bad term. All (practical) optimization problems are constrained, even the seemingly unconstrained ones.

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