Recently, I am trying to solve a stochastic planning problem. The problem is like this:
$$ \max_{x\in \mathbb{R}^n,x\geq0}\quad \varphi(x) = \mathbb{P}[\xi\leq b+Ax] \\ Cx\leq d $$
In this problem, $Cx\leq d$ is linear constraints, which is not a trouble in solving this problem. However, $\xi \in \mathbb{R}^m$ is a Gaussian random vector with variance-covariance matrix $\Sigma$ and mean $\mu$.
As I don't want to find the exact solution to this problem, the maximum objective function is certainly 1. I just want to find an $x$ to make sure the probability larger than a certain value, say 0.98. In other words, I want to formulate this optimization problem as a feasibility problem. However, I don't know how to deal with this objective function which is actually a cumulative multinormal distribution function.
I cannot find any useful material to solve this problem. I would appreciate that if you could give me some suggestion or recommend me some book or paper to read.
Thanks a lot!
This answer had a flaw and the correct answer is that the problem is intractable as pointed out in the comments.