Suppose $X=(X_1, \cdots,X_k)$ follows the multinomial distribution with a known size $n$ and an unknown probability vector $(p_1,\cdots,p_k)$.
Find the necessary conditions for the solution to the maximization of $\prod_{i=1}^n p_i$ subject to the constraints $p_1>0$, $\sum_{i=1}^np_i = 1$ and $\sum_{i=1}^np_ih(X_i) = 0$.
Here's how I deal with it: I need to find the unique Lagrange multipliers $\lambda_*\in\mathbb{R}^c$ that satisfy $Df(x^*) - \lambda_*^T Dg(x^*) = 0$, where $f(x)$ is the objective function and the $g(x)=0$ is the constraint equation. However, I am not sure what are the objective function that I need to take derivative for this case since I am dealing with the maximization of $\prod_{i=1}^n p_i$. What is the correct equation?