Which optimization technique to use with noisy and slow black-box system?
Are you a petroleum engineer by any chance? This sounds very like the sort of optimization you have to perform in oilfield waterflood operations; you have to allocate your limited budget of injection water and/or lift gas to maximize oil production, with the additional complication that the optimal set point changes as the reservoir depletes and/or wells get drilled or abandoned.

The oilfield case has the useful feature that the overall objective function (field oil flowrate) is a simple sum of well flow rates. Each production well is generally only sensitive (in the short term!) to a small local subset of the inputs (gas lift rate in the production well, and water injection rate in near-offset water injection wells).

I would generalise the solution employed in the oilfield as follows: 


Regularly perturb the inputs
Measure the affected local outputs
Maintain a simple linear or low-order polynomial model of the system and its parts
Use the model to optimize day to day
Keep good records to ensure that the model is up to date with reality
Monitor local and global model fidelity to decide what to perturb and recalibrate next


This works because the small perturbations that you keep on making (for model calibration purposes) are of short duration and local effect, so they don't take the overall system very far away from the optimum. The trade-off between keeping the system optimized, and "de-optimizing" it to calibrate the model, works out favorably. The system reacts on a timescale of days to months, and the measurements generally take a day or two to accomplish.

Try Googling waterflood optimization and gaslift optimization for some more detailed pointers, although a lot of the trade literature is paywalled.

Looking forward to learning what approaches have been taken in other fields with this kind of problem.