The phrase "abandon hope" denotes a sense of giving up or losing all optimism, and there are plenty of synonyms that could be used in its place. Despair, relinquish, surrender, resign, and forgo all have a sense of defeat and hopelessness. When one is in a state of helplessness and can no longer push on, they might feel resigned to their fate, completely stuck in their current situation. One might forgo any hope or optimism when it seems impossible to change the circumstances. These synonyms for "abandon hope" all convey that feeling that there's really nothing left to do but accept the bleakness of the situation.