What type of models exist to optimize an output based on certain inputs, if you do not know the mathematical relationship between these, but have access to past input and output data?
For example, let's say I am trying to optimize the purity of a chemical compound in an experiment, and I have done hundreds of experiments in the past, for which I have the output (purity) and inputs (concentration of each component, temperature, pressure, etc…).
Best Answer
If new data is expensive/time-consuming to generate, surrogate-model based optimization may be good approach.
That is:
What defines a suitable model depends on your problem. For something non-linear, neural networks (as noted by usεr11852), support vector machines or Gaussian Process Regression (aka Kriging) might work. On the other hand, the computational effort for training a model will be adversely affected if your data-set is rather large.
Kriging provides the advantage of giving an uncertainty estimate of its prediction. This allows to compute the Expected Improvement of a candidate solution. While this does not exactly "integrate these 2 steps" (modeling+optimization) EI provides the means to elegantly balance Exploitation vs. Exploration in a surrogate-model based optimization framework.
I suggest reading Engineering Design via Surrogate modeling, Forrester et al. (2008). Since you already have pre-existing data, you can probably skip Chapter 1 (sampling plans), and focus on Chapter 2 and 3.
Regarding your constraints: If the constraints themselves are inexpensive to calculate on-the-fly, just respect them in the above step 2. (as you already suggested in your comment). If constraints themselves are expensive to evaluate, you may consider to replace them with a model, too. Similar to EI, you can use the uncertainty estimate of a Kriging model to compute the probability of feasibility.