Maximum CLIQUE problem is very hard to approximate. It has a self-improvement property defined using graph product which is utilized to prove hardness of approximation results. One such example is ruling out any constant factor polynomial time approximation scheme unless $P = NP$. Furthermore, using booster products, we can show that its NP-hard to approximate Max CLIQUE within a factor of $n^{\epsilon}$ for some value $\epsilon >0$. Also, Maximum independent set problem has a self-improvement property which is used to rule out constant factor approximation (assuming $P \ne NP$).

I would like to gain insights into the failure of many optimization problems to have self-improvement property. What are the common features of hard to approximate problems that poses self-improvement property?

Under what conditions an optimization problem can not poses self-improvement property?