Give somebody up for lost
Definition 1 of 1
Pro Tip 1/3Hopelessness Context
Used when all hope is lost for someone's return or survival in dire situations. 
They gave up the lost climbers for dead when the storm hit.
Pro Tip 2/3Not Literal
Can be metaphorical, not always meaning death but perhaps the end of a search. 
After months of no news, she was given up for lost by the investigators.
Pro Tip 3/3Intensity
Expresses a strong emotional response, often sadness or resignation. 
The town gave up the missing dog for lost after the hurricane.