Hill-climbing searches work by beginning with an initial guess of a solution, then iteratively making local changes to it until either the solution is found or the heuristic gets stuck in a local maximum. There are many different ways to try to avoid getting stuck in local maxima, such as running many searches in parallel, or probabilistically choosing the successor state, etc. In many instances, hill-climbing algorithms will rapidly converge on the correct answer. However, none of these approaches are guaranteed to find the optimal solution.
Branch-and-bound solutions work by cutting the search space into pieces, exploring one piece, and then attempting to rule out other parts of the search space based on the information gained during each search. They are guaranteed to find the optimal answer eventually, though doing so might take a long time. For any queries, branch-and-bound based algorithms work quite well, since a small amount of information can rapidly shrink the search space.
For more information regarding Hill-climbing and branch-and-bound algorithms for exact and approximate inference in credal networks, refer to the following link.
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