You can never lose face by winning! Winning isn't everything, it's the only thing.
You can lose by abandoning the very principles you claim to represent. Bush tried to transition Iraq into a democracy before it was ready, so as to bump up flagging support for the war effort. It worked for a while, until it became clear the Iraqi government is both hollow and infiltrated by foreign influence.
If they were to abandon that, our supposed claim to be there, to help the Iraqis build their government, would be revealed to be false. It would all be about preserving American hegemony. That is the truth, but like I said before they painted themselves into a corner pretending it wasn't.
As for the troops not giving up, that's what I expect to hear. There have been other examples throughout history where the troops had the will to win, but the politicians didn't and thus made victory unattainable. Just think of the dedication and patriotism of the Japanese soldiers who were still in a war mode on those tiny Pacific Islands, not realizing Japan had surrendered to the USA decades earlier.