Let $X_i$ be independent, discrete random variables each with a known distribution and $\lambda_i$ be unknown real numbers. I am interested in running a linear optimisation over the $\lambda_i$ optimising for some behaviour of the sum $\sum \lambda_iX_i$.
For the optimisation, I will want to minimise the CVaR (Conditional Value at Risk) subject to some constraints. This is detailed in Uryasev's paper (https://www.ise.ufl.edu/uryasev/files/2011/11/CVaR1_JOR.pdf) and is straightforward if I can approximate the discrete pdf of $\sum \lambda_iX_i$ via a representative sample of $N$ equally likely outcomes.
I can see two approaches:
- Using Monte Carlo sampling to generate $N$ realisations of the $X_i$.
- Using numerical methods to convolve the $\lambda_iX_i$ to identify the most representative $N$ realisations of $\sum \lambda_iX_i = f(\lambda_i)$. Exploiting the indepdendence of the $X_i$ in this specific problem.
Option 1 is mathematically straightforward but I am concerned about getting a representative sampling. I would appreciate any literature on effective sampling techniques to ensure convergence (particularly since the weighting of the $X_i$ is a variable).
I am not sure how to make further progress on option 2. I have explored the Fast Fourier Transform which will allow me to find the convolution for any fixed $\lambda_i$ but am unsure if this will generalise to variable $\lambda_i$. Any guidance on whether or not this is feasible would also be appreciated.