  We discuss investment allocation to multiple alpha streams traded on the same
execution platform with internal crossing of trades and point out differences
with allocating investment when alpha streams are traded on separate execution
platforms with no crossing. First, in the latter case allocation weights are
non-negative, while in the former case they can be negative. Second, the
effects of both linear and nonlinear (impact) costs are different in these two
cases due to turnover reduction when the trades are crossed. Third, the
turnover reduction depends on the universe of traded alpha streams, so if some
alpha streams have zero allocations, turnover reduction needs to be recomputed,
hence an iterative procedure. We discuss an algorithm for finding allocation
weights with crossing and linear costs. We also discuss a simple approximation
when nonlinear costs are added, making the allocation problem tractable while
still capturing nonlinear portfolio capacity bound effects. We also define
"regression with costs" as a limit of optimization with costs, useful in
often-occurring cases with singular alpha covariance matrix.
