This is motivated by my previous question, Super-polynomial time approximation algorithms for MAX-3SAT. For many optimization problems, for each one we have inapproximability lower bound $\alpha$ assuming some widely believed complexity theoretic conjecture. In other words, there is no polynomial time algorithm for such optimization problems with approximation ratio better than some $\alpha$ (different ratio $\alpha$ for each problem).
Are there optimization problems for which we can achieve approximation ratio better than $\alpha $ if we allow super-polynomial time algorithms? Can we achieve better approximation ratios using quasi-polynomial time algorithms ( $n^{O(\log n)}$) or even using sub-exponential time algorithms($2^{o(n)}$)?
I would appreciate a survey of such results.