Reduction from SAT to 0,1 integer linear program with zero or one solutions
OptP-complete. Krentel showed that MAX-SAT, finding the lexicographically maximum satisfying assignment, is OptP-complete and the reduction above reduces Max-SAT to ILP. ILP sits in OptP pretty much by definition.

Note that you need n calls to an NP-oracle to solve ILP via binary search, O(log n) isn't sufficient. 

There really isn't much of a connection to UP, though Mulmuley, Vazirani and Vazirani give an isolation lemma that gives an alternate proof of Valiant-Vazirani via a maximization problem.