Parametrized linear optimization problem

Let $ p_0, p_1, \ldots p_{L-1} $ be L different probability distributions over M elements:

$ p_i = (p_i^0, p_i^1, \ldots p_i^{M-1}) $ where $ p_i^j \in (0,1), \forall i=0\ldots L-1, \forall j=0\ldots M-1 $, and$  \sum_{j=0}^{M-1} p_i^j = 1, \forall i = 0\ldots L-1  $.

Let $ c_0, c_1, \ldots c_{L-1} $ be L real, positive constants. Let $ a $ be a real positive constant, and let $ \delta \in (0,1) $.

We have the following linear optimization problem:

Maximize $ f  =\delta p_0 x $
Such that:
- $ x^T = (x_0,\ldots, x_{M-1}) $ are M positive real variables;

- $ x_i - \delta p_0 x \leq (1-\delta) a, \forall i= 0,\ldots, M-1; $

- $ \delta (p_j - p_0) x \leq (1-\delta)(c_j - c_0), \forall j=1,\ldots,L-1; $

We know that for some value of $ \delta = \delta^* $, the optimization problem accepts a solution with the maximum objective $ f = f^* $. I would like to know the minimum value of $ \delta $ such that the problem still accepts a solution.

Thank you for your help.

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