$\mathbf{Q}:$ Suppose we have the following chance constrained optimization problem:
\begin{aligned} & \text{minimize} & & x_1 +x_2\\ & \text{s.t.} & & \mathbf{P}(x_1 + 2x_2\geq\xi_1) \geq\alpha_1\\ &&& \mathbf{P}(2x_1 + x_2\geq\xi_2) \geq\alpha_2\\ &&& x_1\geq 0,x_2\geq0 \end{aligned}
where $\xi_1$ and $\xi_2$ are nonnegative random variables which are independent and continuously distributed with a probability distribution function $F_1$ and $F_2$ respectively. Find the optimal solutions $(x_1^*(\alpha_1,\alpha_2),x_2^*(\alpha_1,\alpha_2))$ for any $(\alpha_1,\alpha_2)\in [0,1]\times[0,1]$.
$\mathbf{A}:$ My approach is as follows:
Since the RHS of the constraints are of the form $h=\binom{\xi_1}{\xi_2}$ which is random, the feasible sets of the chance constraints are thus respectively, $K(\alpha_1)=\{(x_1,x_2)|x_1+2x_2\geq F_1^{-1}(\alpha_1)\}$ and $K(\alpha_2)=\{(x_1,x_2)|2x_1+x_2\geq F_2^{-1}(\alpha_2)\}.$
Am I right at till this point with regards to my approach? I am clueless as to how I can proceed further, and I tried simplex but got stuck as well..
Some help will be deeply appreciated!