Suppose the function I wish to optimize is $f(x)$, which is twice differentiable and concave. Suppose that, as part of the problem, with probability $p_1$, the constraint is $c_1(x) = 0$, and with probability $p_2$, the constraint is $c_2(x) = 0$. Is the correct way to set up the Lagrangian in this case
$$f(x) - p_1 \lambda_1 c_1(x) - p_2 \lambda_2 c_2(x)?$$
Is there any work on Lagrangians with probabilistic constraints, as described here?