That would be a solution where some variables are fixed (their values are known) and some other are still to determine. The partial solutions are associated with the nodes of the branching tree that are not leaves (as Samuel reminded me, except for the nodes that became leaves by pruning, these are partial solutions too). For example, if you're optimizing a function over a subset of R^4, an example of a partial solution would be (1,x2,2,x4), x2 and x4 being subject to some set of constraints.
I hope my answer helped.
Best regards,
Imène.
During the process, a partial solution X might finally become a leaf (by pruning) even if all the variables aren't integral. Indeed, you could find an integer solution which gives a better objective values than X. Therefore, it's useless to search in the X branch.
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