## Parametrized linear optimization problem

Let be L different probability distributions over M elements:

where , and.

Let be L real, positive constants. Let be a real positive constant, and let .

We have the following linear optimization problem:

Maximize
Such that:
- are M positive real variables;

-

-

We know that for some value of , the optimization problem accepts a solution with the maximum objective . I would like to know the minimum value of such that the problem still accepts a solution.