This is great. The pragmatic bit that makes it a bit more challenging (or, perhaps, a bit more of a search optimization problem than analytical) is that most of the cost functions are step functions. Once you need better than 10% tolerance on a resistor, you might as well step down to 1% -- available tolerances are discrete, and 5% and 2% tolerance parts are rarely cheaper than 1%. Similarly going to 0.1%. So once it's clear that preferred_tolerance_N is not going to solve, you can assume preferred_tolerance_N+1, and then push that through -- and hopefully push a few marginal parts up a bucket. There's also a similar step-function cost for adding a new BOM line, especially for hobby-scale projects; this is a project-global optimization, where I might change a 10 kΩ / 2.2 kΩ resistor ladder to 6.8 kΩ / 1.5 kΩ if I already have 1.5 kΩ, 6.8 kΩ, and 10 kΩ in my BOM but would otherwise need to add 2.2 kΩ (assuming everything else still solves, of course; more likely than not, since we usually don't care much about the total resistance of a ladder, as long as it's stiff enough, and doesn't leak too much).
Hadn't thought about the BOM line one but that definitely makes sense. The implementation I linked is definitely math heavy but might actually not be the best idea. As you say, the cost functions has steps. For less than a given amount of components, random search might be suited. Past that point, we might use maths to constrain the problem and then search within that range.
We have equations at the top of our todo list and hope to get there soon :) Matt setup a roadmap here with things we want to work on: https://atopile.io/roadmap/
