How can I constrain a variable in a linear program to be in a given distribution? given a standard linear program where
$max$ $c^T.v$
$s.t.$
$s.v=b$
$v_l<v<v_u$
for a given variable $v_i$, in addition to the lower and upper bounds, I know the 
probability distribution that $v_i$ follows (say a normal distribution).
Is there a way to add this information in the linear program, in a way that penalizes lower probability values (tails) and favors higher probability values (mean).
I found items linked to probabilistic programming but wasn´t sure if it was the answer.
Thanks.
 A: What I have seen to solve this kind of problem is called stochastic programming (but probabilistic programming seems to be also a valid name for the same thing).
However, something I cannot see is if you $v_i$ of which you talk about is part of the data or a variable (you said it was a variable). 
If it is part of the data, then stochastic (probabilistic?) programming applies. In that case, some changes occur in your program, and you have some tuning to do depending on how much you can afford to violate your constraints. You will also have to choose between maximising the expectancy of your cost function, or minimizing the regret, or other stuff.
If it is a variable, then you can choose its value. If your probability is a probability that $v_i$ is equal to some value in the optimal solution, then you won't need any stochastic programming. If $v_i$ is integer, you can take this into account in the branching strategy of your branch and bound (you know where to look for). If it is real, you can try to tune your simplex (or other local descent) algorithm so that it looks for the most likely solution, but I don't think it will be useful, since it just seems faster to look for the local optimum.
