PH and Optimization Problems
Let machine M solves the $\Sigma_2^p$-complete problem. We can use  this  machine to solve 
any "NP-Hard" optimization problem (which includes MAX-CLIQUE and MIN-COLORING)  in 
additional time polynomial in size of input as follows:

The solution is described for MAX-CLIQUE but the same idea works for any NP-Hard optimization problem. Let G = (V, E) and

O = { $&lt;G,k&gt;$ : the size of largest clique in G is k }

D = { $&lt;G,k&gt;$ : there is a clique of size >= k in G  }

Observation 1: Problem of deciding language O can be solved by making $\log{|V|}$ calls 
to  the machine that decides D [Using Binary search]


Reduce instance of O to instance of D by using observation 1
Reduce instance of D obtained to instance of $\sigma_2^p$ complete problem P
Solve P using M


The reductions are all polynomial time, so its easy to see that if M works in polynomial
time then we can compute MAX-CLIQUE using M with polynomial time overhead.

Using the above reduction we can also see that M is at least as powerful as the machine
that solves NP-hard optimization problems.