Correlated Equilibrium - Transforming a non-linear objective function into a linear one

I am trying to transform a non-linear objective function into a linear one, in order to create a LP. How might I go about to do this (I have never taken a course in linear programming).

I have that I would like to maximize the following objective function: $\mu(\alpha)$, where $\mu(\alpha) = \min{U_i(\alpha)}$. $U_i(\alpha)$ denotes the expected utility for player $i$ in a correlated Nash equilibrium and $\alpha$ denotes the corresponding probability distribution played by the players.

The contrains I have so far are the following: $\sum_{a \in A} \alpha(a) = 1$, and $\alpha(a) \ge 0$ for all $a \in A$. The other contraints that I have are the ones used for a standard correlated equilibrium, namely that: for each player $i$ and pairs of action $a_i^{1}$ and $a_i^{2}$, we have that: $\sum_{a \in A | a_i = a_i^{1} } [u_i(a) - u_i(a_i^{2}, a_{-i})]\alpha(a) \ge 0$.

The first 2 contraints are the ones requiring that $\alpha$ follows a proper probability distribution, and while the last sets are the ones that enforce the correlated equilibrium (by definition). How might I go about to modify this into an LP with the objective function I mentioned above. I know that it's not linear, so I need to change it somehow (and the LP ) appropriately.