  We consider the classical problem of optimal portfolio construction with the
constraint that no short position is allowed, or equivalently the valid
equilibria of multispecies Lotka-Volterra equations with self-regulation in the
special case where the interaction matrix is of unit rank, corresponding to
species competing for a common resource. We compute the average number of
solutions and show that its logarithm grows as $N^\alpha$, where $N$ is the
number of assets or species and $\alpha \leq 2/3$ depends on the interaction
matrix distribution. We conjecture that the most likely number of solutions is
much smaller and related to the typical sparsity $m(N)$ of the solutions, which
we compute explicitly. We also find that the solution landscape is similar to
that of spin-glasses, i.e. very different configurations are quasi-degenerate.
Correspondingly, "disorder chaos" is also present in our problem. We discuss
the consequence of such a property for portfolio construction and ecologies,
and question the meaning of rational decisions when there is a very large
number "satisficing" solutions.
