If solving the whole belief space exactly is hopeless, don't try. Compute good responses for a representative handful of belief-states ahead of time — a pre-built playbook keyed to how sure you are the project is in trouble.
Offline belief-state planning computes an approximate policy before execution, so that at run time you simply look up the response for your current belief. The breakthrough is to stop insisting on the value of every belief and instead pin it down only where it matters — at a set of sampled, representative beliefs you're realistically likely to encounter.
1Point-based value iteration
The workhorse family is point-based value iteration. Rather than maintaining alpha vectors across the whole continuous belief space, it samples a finite set of reachable beliefs and maintains one alpha vector anchored at each. Backing up value only at those points keeps the computation bounded while still building a piecewise-linear approximation to the optimal value function. Refined versions concentrate their sampled beliefs on the regions the optimal policy actually visits, getting excellent policies for surprisingly large problems.
2Cheaper bounds and shortcuts
Simpler approximations trade more accuracy for speed. The QMDP shortcut pretends the state will become fully observable after the next step — fast to compute, but it systematically undervalues gathering information, so it can be reckless about acting to reduce uncertainty. Other bounds and grid-based schemes offer different points on the accuracy-versus-effort curve. The right choice depends on how much your decisions actually hinge on resolving uncertainty.
You cannot pin down the value of every belief, so pin it down only at the beliefs you'll actually face, computed in advance. A policy that's near-optimal on the belief-states your projects really reach beats an exact one you can never compute.
This is contingency planning made rigorous. Rather than improvising when a project's status turns ambiguous, work out your best responses ahead of time for a representative set of situations — "fairly sure it's fine", "genuinely uncertain", "fairly sure it's in trouble" — and concentrate that effort on the states your projects actually tend to reach, not every theoretical possibility. And be wary of shortcuts that assume uncertainty will conveniently resolve itself: they'll under-invest in the very information-gathering that de-risks a project.