Solving convex combination of Arithmetic and Geometric on random variables This is my term project for non-linear stochastic optimization and the main problem is solving $f(x_{1},...x_{n})$, i.e., I need to maximize $f(x_{1},...x_{n})$ and find $X=(x_{1},...x_{n})$
where $0\leq z \leq 1$ is known, $0\leq x_{i}\leq 1$ and $\sum_{i=1}^{i=n}x_{i} =1$.
$Y_{1},...Y_{n}$ are independent positive (>0) random variables with probability distribution $P_{Y_{i}}$ and mean $\mu_{i}$ (expected vlaue :$\mu_{i}$ =$E[Y_{i}]=\sum P_{Y_{i}}Y_{i} $).
Also, the probability distribtion of the convex combination of $Y_{i}s$ ,$\sum_{i=1}^{i=n}x_{i}*Y_{i}$, is  $P_{Y1,...,Yn}$. Due to indepency of random variables $P_{Y1,...,Yn}=P_{Y_{1}}*P_{Y_{2}}*...P_{Y_{n}}$.  
The function f is a convex combination of Arithmetic-Geometric (A-G) means, $z$ handles the convex combination of A-G means. 
$\sum_{i=1}^{i=n}(x_{i}\mu_{i})=\sum_{i=1}^{i=n}(x_{i}E(Y_{i}))$ is the arithmetic mean of random variables 
and the second term is geometric mean of the random variables that I converted to $exp(.)$ term to make it easier to deal with. In original form, it should be $\Pi(\sum_{i=1}^{i=n}(x_{i}Y_{i}))^{P_{Y_{1},...,Y_{n}}}$
= $exp(P_{Y_{1},...,Y_{n}}\ln(\sum_{i=1}^{i=n}(x_{i}Y_{i})))$. in other words, its $E[\ln(\sum_{i=1}^{i=n}(x_{i}Y_{i}))]$.
In this problem everything is known expect $X=(x_{1},...,x_{n})$, and
I need a closed form solution based on $X$ but due to the nonlinearity of the function $f$, I believe some estimations such as Taylor is needed to get rid of $ln(.)$ and/or $exp(.)$ terms.
As an simple example, let assume $n=2$, $P_{Y_{1}}(Y_{1}=2)=1$, and $P_{Y_{2}}(Y_{2}=1)=P_{Y_{2}}(Y_{2}=3)=0.5$ and $E[Y_{1}]=1, E[Y_{2}]=2 $ and $z=0.5$, then $f(x_{1},x_{2})$ is 
$f(x_{1},x_{2}) =0.5(2x_{1}+2x_{2})+0.5*\exp(0.5\ln(2x_{1}+x_{2})+0.5\ln(2x_{1}+3x_{2}))$. now, the question is finding $X=(x_{1},x_{2})$ where $ 0  \leq x_{1},x_{2}\leq 1$ and $x_{1}+x_{2}=1$. 
 A: Your example is trivially solved with optimal $x_1 = 1, x_2 = 0$, regardless of the value of $z$ (irrelevance of the value of $z$ in this example is because means of the random variables are equal). In this particular example, $f$ is monotonically increasing from its minimum at $x1 = 0$ to its maximum at $x_1 = 1$.  
More generally, a problem of the form you provided will be a linearly constrained nonlinear optimization problem which can be solved using a numerical nonlinear optimizer which can accept linear constraints.  
If there's a general closed form solution, I'll leave that for someone else to figure out. But perhaps there is a closed form for special cases.  It's easy to calculate the gradient of $f$, and therefore easy to write out the Karush-Kuhn-Tucker conditions https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions which due to the linear constraints, are necessary for a maximum, but are sufficient for a maximum only if $f$ is concave on the constraint region. However, I don't think they will admit a closed-form solution, except in cases, such as your example, in which the optimum is at a vertex. Perhaps someone can characterize the conditions on the probability distributions for which the solution will be at a vertex.
Edit: Why do you "need" a closed form solution? What will the closed form solution be used for? If the closed form solution is an approximation, does it matter how accurate the approximation is? I presume it does, in which case the approximation could come at an inaccuracy cost which is unacceptable. Is a solution obtained by numerical nonlinear optimization acceptable? If not, why not?
