I am just wondering how offensive would it be to promote my open source project via LinkedIn messages? I am thinking about buying a premium access in LinkedIn so I can search people who might be interested in my project and sending them mail However I am not sure it is ethical or how annoying it can be for them.
Advertising for open-source software is no different from advertising other products or services you might offer. I am calling it advertising, because that is what your messages would be doing: bringing your product to the attention of others.
Sending unsolicited messages, regardless of the medium, is spamming and as such is not appreciated and might even be illegal in some jurisdictions.
A better approach would be to publicly post about your project, so that the interested people can read about it on their timeline. However, be careful you don't overdo it, both in frequency of posting and the tone of your posts, or people will just start to ignore your messages.