  We consider the issue of solution uniqueness for portfolio optimization
problem and its inverse for asset returns with a finite number of possible
scenarios. The risk is assessed by deviation measures introduced by
[Rockafellar et al., Mathematical Programming, Ser. B, 108 (2006), pp. 515-540]
instead of variance as in the Markowitz optimization problem. We prove that in
general one can expect uniqueness neither in forward nor in inverse problems.
We discuss consequences of that non-uniqueness for several problems in risk
analysis and portfolio optimization, including capital allocation, risk
sharing, cooperative investment, and the Black-Litterman methodology. In all
cases, the issue with non-uniqueness is closely related to the fact that
subgradient of a convex function is non-unique at the points of
non-differentiability. We suggest methodology to resolve this issue by
identifying a unique "special" subgradient satisfying some natural axioms. This
"special" subgradient happens to be the Stainer point of the subdifferential
set.
