Solutions
A solution to a problem solves the problem to your satisfaction. Solutions don't have to be perfect, and you may want a new solution later.
When multiple people are involved, a solution solves the problem to everyone's satisfaction.
A compromise is not a solution because people give up part of what they want and aren't fully satisfied with the outcome. Also, a compromise is not a way of proceeding that any person says is the best thing to do; how can you expect an idea with no advocates at all to work out well?
When people have contradictory ideas, there are two possible ways to achieve a solution:
1) Base the solution on something you do agree about. At a minimum, you may be able to agree to go your separate ways.
2) At least one person, and usually everyone, will change their mind (possibly only in a small way).
Voluntarily changing one's mind is a purely good thing. It means one has a new idea one considers better than the old one. There is no downside whatsoever.
Some people are scared of changing their mind because they fear they won't get what they want. If they actually changed their mind, they wouldn't miss whatever they weren't getting because they wouldn't want it anymore. If they still want it, they haven't actually changed their mind. Pressuring people to change their mind is good at getting them to say they have, but bad at getting them to actually do it, and can lead to this kind of confusion.
The way to achieve mind changing is with persuasion and brainstorming. That means giving good explanations that help others understand what you're getting at, and it means coming up with new ideas that aim to be better than anyone's initial ideas. Persuasion also makes use of criticism: if I understand the flaws in an idea, then that gives me a good reason to adopt a new idea.
Put another way, one can change his mind by creating knowledge of the new ideas to have, of the flaws in the old ideas, and of how to transition between them. By tying the issue to knowledge creation, we can see that the way forward is with guesses and criticism.
Example
Suppose I want a cat, and my roommate wants a dog. Neither of us wants multiple pets. We have a problem. She might suggest we get an owl instead, and point out the merits of owls. If we both find this more appealing than our original ideas, it could be a solution. Then I point out that dogs require more chores than cats. You have to walk them and their feces are larger. This may contribute to a solution by giving my roommate a reason to be happy about not having a dog. If my roommate points out that cats don't come when they are called, and can't be trained like dogs, that isn't being mean, it's helping me see some reasons I might not want a cat after all. It's telling me some merits of dogs, and demerits of cats, I may have missed. It's completely up to me though. If I still want a cat, so be it, no one is forcing me to change my mind, they're just offering ideas that might be relevant or helpful. Next I suggest that it might be time for me to move out and get my own place, and then we could get separate pets. But we decide we don't want to do that. Next, we decide to watch videos depicting cat and dog ownership. After watching for a while, we realize it's easier and more fun to watch highlight clips of other people's pets than to take care of our own. So we have a solution that we both prefer to our original ideas and we're both happy with.
This solution may be temporary. Maybe next week I'll think of a new reason I do want my own pet. Or maybe I'll just feel a nagging doubt and that unpleasant emotion will be a problem to solve, possibly by getting a pet. But for now it's a solution, and it might turn out that no further problem ever comes up.