Negotiation with difficult or unreasonable parties

I’m stuck on a problem…I’ve been involved in long, ongoing negotiations regarding my business’ primary source of income. I have a valid contract in place, but the customer wants to change the terms of our business in ways that would add substantial cost and risk to my business. I have been amenable to considering many of the changes, and have crafted multiple solutions that would get us both what we need, but all have been rejected.

The customer does not want to compromise or “give” anything to get us both to “yes”, and I keep going back to the drawing board to come up with new solutions because I do want to help and ideally want to keep their business – I’m just not willing to do it without some consideration for the huge cost and risk involved, especially when we have an existing contract in force.

I have tried repeatedly to show the value in the proposed solutions (and it’s there), but the customer is conceptually/emotionally opposed to anything but their own solution and keeps holding out, to their own detriment! How do you get through to unreasonable people to reach a compromise?