My partner violated my boundaries in a newly-open relationship

My partner and I have recently opened our relationship. Because I am currently prioritizing my mental health, I have asked only one thing: please do not let me see your relationships. Don't Ask Don't Tell is not my ideal scenario, but for the moment it is what I need. My partner agreed, but he continually pushes and even breaks that boundary. In one case, he said his new partner wanted to be publicly acknowledged, so he chose her comfort over mine. I feel violated and disregarded. My heart hurts.

You have the right to set the boundaries that you need, but other people aren't required to do everything you ask if they are unable to. In that case, it's their responsibility to say "I'm sorry, I can't meet that need/stay within that boundary" - it's pretty crappy to make a promise and then keep breaking it.

The issue here is that you asked your partner to abide by specific rules for opening the relationship, and he is not abiding by those rules. You can't force him to change his behavior - you can just decide how to respond.

You can say "I'm actually not comfortable having an open relationship now that we've tried it and learned that your way of practicing non-monogamy isn't compatible with my needs. We tried it, it's not working for me, and I am rescinding consent to open the relationship." If he 'refuses,' then the relationship is over - if you want to be monogamous, and he doesn't, then that's a serious impasse.

You can also decide that you're already done, if him pushing at your boundaries this way is a dealbreaker for you. (It would be for some people; it wouldn't be for other people.) You can say "You've demonstrated that you're willing to ignore, disrespect, or fudge boundaries with me, and that makes me feel uncomfortable dating you now that I know that." 

Or, you could try and dig down to the root of your need for DADT right now, and try to work something out with your partner that recognizes his need for relationships in the open while also helps you manage your mental health. Consider finding a poly-friendly therapist to talk to about this, and set some treatment goals. Sit down with your partner and talk about what needs, fears, and desires you're trying to address with your DADT request, and what needs, fears, and desires make this so hard for him. There might be a way to learn and heal and move through this to an arrangement that's healthy for everyone. But you're not obligated to do that work; it's always your right to leave a relationship that's just not working for you.