If your objective function is piecewise linear and concave, then it is the minimum of a bunch of globally linear functions. Let me put this in the more usual minimization framework (instead of maximization -- just flip the sign):

So you're trying to solve $$ \min_\lambda f(\lambda) $$ where $f(\lambda)$ is piecewise linear and convex. Then I can write $$ f(\lambda) = \min_i f_i(\lambda) $$ where each of the $f_i$ are linear. There is a canonical way of solving this, namely by introducing a scalar slack variable $\mu$ and writing the problem as $$ \min_{\lambda,\mu} \mu \\ \text{so that } \mu \ge \min_i f_i(\lambda), \qquad i=1...N. $$ This is now a linear program: the objective function and all constraints are linear. Even if large, there are very efficient ways of solving this problem if only you can characterize the $f_i$. In your case, I imagine these are exactly the $E_\sigma(\lambda)$ for each of the possible $\sigma$.

If your objective function is piecewise linear and concave, then it is the minimum of a bunch of globally linear functions. Let me put this in the more usual minimization framework (instead of maximization -- just flip the sign):

So you're trying to solve $$ \min_\lambda f(\lambda) $$ where $f(\lambda)$ is piecewise linear and convex. Then I can write $$ f(\lambda) = \min_i f_i(\lambda) $$ where each of the $f_i$ are linear. There is a canonical way of solving this, namely by introducing a scalar slack variable $\mu$ and writing the problem as $$ \min_{\lambda,\mu} \mu \\ \text{so that }\quad \mu \ge f_i(\lambda), \qquad i=1...N. $$ This is now a linear program: the objective function and all constraints are linear. Even if large, there are very efficient ways of solving this problem if only you can characterize the $f_i$. In your case, I imagine these are exactly the $E_\sigma(\lambda)$ for each of the possible $\sigma$.