  This paper considers stochastic optimization problems whose objective
functions involve powers of random variables. For example, consider the classic
Stochastic lp Load Balancing Problem (SLBp): There are $m$ machines and $n$
jobs, and known independent random variables $Y_{ij}$ decribe the load incurred
on machine $i$ if we assign job $j$ to it. The goal is to assign each jobs to
machines in order to minimize the expected $l_p$-norm of the total load on the
machines. While convex relaxations represent one of the most powerful
algorithmic tools, in problems such as SLBp the main difficulty is to capture
the objective function in a way that only depends on each random variable
separately.
  We show how to capture $p$-power-type objectives in such separable way by
using the $L$-function method, introduced by Lata{\l}a to relate the moment of
sums of random variables to the individual marginals. We show how this quickly
leads to a constant-factor approximation for very general subset selection
problem with $p$-moment objective.
  Moreover, we give a constant-factor approximation for SLBp, improving on the
recent $O(p/\ln p)$-approximation of [Gupta et al., SODA 18]. Here the
application of the method is much more involved. In particular, we need to
sharply connect the expected $l_p$-norm of a random vector with the $p$-moments
of its marginals (machine loads), taking into account simultaneously the
different scales of the loads that are incurred by an unknown assignment.
