Desperate is when everything depends on something succeeding. 'He was desperate to find his child', 'He made a desperate attempt to save his business'. Optimism is when you look at a situation from a positive or hopeful viewpoint. So - desperate optimism is when you look at something positively, when that thing has to succeed. Desperate optimism could be used to describe someone who was positive about the outcome of a cancer treatment. You could also use it to describe a businessman's last attempt to save his business before he goes bankrupt. You can obviously be desperate without being optimistic - and be resigned to your fate.