In general, solving an optimization problem is very hard. However it is never well explained why these types of problems are difficult to solve from a computational complexity and theoretical computer science point of view.
In this project, I presented a lecture in which I explained the hardness of approximation using Probabilistic Turing Machines. Specifically I coverd the topic of PCP theorem, a strong result in complexity theory, that links NP class complexity with a special complexity class called PCP.
PCP thorem implies a lot of consequences in several fields of mathematics and discrete optimization.
In presentation.pdf you can see the presentation.