minimize the cost to taxpayers

You can of course do that, or any number of other things where you pick a metric and optimize it. The question is: how well does that metric capture what we actually want? For me, at least, optimizing (min wage * #jobs) doesn't seem like it matches my values terribly well, though it's probably better than maximizing either factor on its own.

a limited number of sweatshops

That's an interesting idea (though it feels rather horrible), but I'm not sure how it's supposed to work.

• If they are meant to be the only safety net for people who can't find work at minimum wage or above: why would we expect it to be sufficient? Some people may be unable to cope with working conditions in the sweatshops; some people may simply not be able to do the work; and if the amount of sweatshop work is limited as you propose, there's no reason why there should be enough for all the people unable to find minimum-wage-or-better work.
• If there is meant to be some further safety net: why then would anyone work in the sweatshops? The usual answer would be something like "because people like to work; it gives them more sense of dignity and purpose", and indeed people do mostly like to work rather than depend on government benefits. But now we're talking about working conditions and pay that are almost illegally bad, so much so that the government doesn't allow more than a very limited number of people to be stuck with them; I would expect a lot of people to prefer depending on government handouts to that, and I don't think I'd blame them.
  • So maybe they'd be some kind of coercion? You don't get your government benefits if you refuse a sweatshop job, or something. But now (1) this is not reasonable for people who, e.g., are physically incapable of doing that work, and (2) since sweatshop places are scarce, it means that some benefit claimants will (arbitrarily?) be required to work in the sweatshops and some won't, which will surely cause resentment. I suppose you could require everyone in receipt of benefits to work in the sweatshops some fraction of the time?