I should clarify - it's at the start of runtime when the library is initially loaded. Not afterwards. You'd have to restart the program to swap a library.

For testing and simulation, it's a way to mock dependencies. You get to test the otherwise exact binary you'll deploy. And you can inject tracing and logging.

Solver installations tend to be long-lived, but require customizations, upgrades, and fixes over time. Dynamic linking lets you do that without going through the pain of reinstalling it every time. Also with solvers you're likely to see unconventional stuff for performance or even just due to age.