If we think about it, we don’t normally have a fixed budget for most purchases. We have a certain utility we expect to derive from them and we hope to spend as little as possible on them (but we don’t have a maximum budget).
In this case, it is utility that is fixed as a restriction, and cost that we can play around with. It’s like solving the consumer’s choice problem in a mirror image way to utility maximisation, and is associated with Hicksian demand curves.
The way we resolve this minimisation problem is very similar to utility maximisation, and is also done with a Lagrangian system, since there is a duality in consumption.