give someone up for lost
give (someone or something) up for lost
To stop pursuing or searching for someone or something because it seems like a futile task. I haven't been able to find that watch for weeks, so I've given it up for lost at this point.
Farlex Dictionary of Idioms. © 2015 Farlex, Inc, all rights reserved.
give someone up for loststop expecting that a missing person will be found alive.
Farlex Partner Idioms Dictionary © Farlex 2017