Dating an individual with a dismissive-avoidant attachment style is not impossible, but it is challenging (especially for an anxious-attacher aka an individual with an anxious-preoccupied attachment style). 

A healthy relationship is possible when both parties understand their own attachment style going into the relationship, are actively working on themselves and can express and communicate their needs in healthy ways to one another.

A healthy relationship is not possible however when either or both persons don't know their attachment style AND are an anxious and avoidant pairing because subconsciously and at a core level, the anxious person is so desperate for love, intimacy and connection while the avoidant is also longing for love and security in a relationship but is so afraid of rejection and pain that they have a great tendency to avoid getting too close altogether.

If an anxious and avoidant are in relationship together ​this is when they begin the push and pull dance of intimacy. This is a rollercoaster of emotions mixed with protest behaviors and insecurities from the anxious-attacher and distancing and dismissing from the avoidant.

If you are an anxious-attacher, know this: we have a tendency to want the relationship to move faster into love and deep feelings in the dating phase than an avoidant would. It's not up to the anxious person to convince, persuade or push the avoidant into loving them faster or to comitting to the relationship sooner.

Even in the very early days in the dating phase of a relationship there's a difference in how an anxious verus an avoidant wishes to connect. Where an anxious person would like to talk to and connect with their dating partner every day, if not throughout the day, the avoidant-attacher would be easily comfortable with every couple of days to few days.

This can set alarm bells off for the anxious-attacher, making them think that this person is no longer into them, or that they're seeing other people. Even though it may not be the case! This is when we begin to push them for more connection; expressing in subtle or overt ways that we feel the connection has changed, that they have changed and we demand to know and understand why.

Unfortunately this reaction does the exact opposite of what we want-- it pushes our partner away.

The more we push, the more they pull away. And so begins the cycle.

What is our role here in an anxious-avoidant relationship if you find yourself in one?

First, ask yourself:

Does this person have the ability and will to work on themselves and their own fears of intimacy?
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How can I express my need for closeness and connection in a way that still respects their need for space and time to warm up in the relationship (I'm referring to if these two are in the dating phase)?

Can I provide this space and patience that this person needs to warm up to the relationship and build their feelings in their time (and not mine) while still taking care of my own needs? 

It's very important here to be super clear with yourself on what are and are not willing to tolerate, especially at the beginning of a new relationship. You must first and foremost honor yourself.