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I am trying to minimize the following function :

$$J(p) = JSD(p_u || p)$$
with constraints :
$$ int p = 1 $$
$$ p(x) geq hat{pi} p_p(x) $$
where $JSD$ is the Jensen Shannon Divergence, $p_u = pi p_p + (1-pi) p_n$, $hat{pi} > pi$, and $p_p$ and $p_n$ are probability distributions with disjoint supports.

I formulated the problem as a minimization problem with constraints to try to solve it with a Lagrangian, but I don't really know how (and if) this works on a functional space.

edit:
Let L be the Lagrangian: $$L(p,lambda,mu) = JSD(p_u||p)-lambda int p -1 - int_x mu(x) (hat{pi}p_p - p)(x)dx$$

Functional derivative yields:
$$frac{partial L}{partial p} = ln(frac{2p}{p+p_u})-lambda+mu p = 0$$
After a discussion on the value of $mu$ (0 or not), I show that as I suspected :

On $supp(p_p)$, $p(x) = hat{pi}p_p$

On $supp(p_n)$, $p(x) = (1-hat{pi})p_n$

Two questions remains :

1. How to prove rigorously the necessary condition on the minimum (with the Lagrangian...) ?




2. How to prove there is a global minimum ? (sufficient condition)