Portfolio management is about solving problems in the real world.
In the real world, it is highly unlikely that EVERY asset has a negative expected return. If all the assets in your universe have negative returns, expand your universe to include a short-term fixed income security that is bound to produce a return greater than (or at a minimum equal to) zero. Alternately, assume it is a recession and every asset has a negative return over a short horizon. Even then, it is highly unlikely that the negative return is identical for all. This would suggest that some long-only managers could lose less money by focusing on the securities that are likely to fall less in a recession. Long-short managers could go short some of the other securities with even weaker expected returns.
There's no mathematical reason you can't have a budget constrained between 0 and 1, even if you don't consider a risk-free or short-term fixed income instrument. Then, the optimal portfolio would invest in nothing. In the real world, budget constraints are driven more by business reasons than any mathematical reason. Investors don't want to pay equity (or hedge fund) manager fees for investment management companies to not invest their money in anything. Further, many managers are not judged on an absolute basis, but a relative one. What matters is what their benchmark is doing or what their peers are doing. So even if the benchmark declines, they can still make large institutional managers happy if they decline by less than the benchmark.