I am teaching myself branch and bound and keep coming across the term optimality gap. So I am looking at an example (shown in the image below) and would like to know if my analysis is correct. I think the arrows drawn in the example are misleading me but I have found a way for it to make sense to me and I would like to confirm that I am correct. Note the top value in the node circles are the local upper bound and the bottom value is the local lower bound.

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So based on the image above, the following is my analysis:

  1. Starting at N1, no upper bound has been determined and the global lb at the root node is -13/2 (not shown on the diagram).

  2. After branching on the root node, the global ub remains unchanged but the now the global lb is given as the min(-13/3, -16/3) = -16/3. Min is used as this is a minimization problem.

  3. Ignoring the blue arrow and branching on N2, the global ub now becomes -3 due to a feasible optimal solution being found. Furthermore, since N5 is infeasible and no solution exists, the global lb is now given by the min(-3, -16/3) i.e. a comparison between the local lb of N4 and N3. Hence a global lb of -16/3.

  4. Now branching on N3, we get N6 and N7, however there is no solution associated with N7 as it is infeasible and the global lb is now -22/5 as this is the only active node left. Also the global ub remains as -3 as no new feasible solution is found.

  5. Lastly branching on N6, we get N8 and N9, the new global lb is given as the min(-4, -3) = -4. Since both are feasible solutions, the nodes are fathomed and the global ub is updated as a better feasible optimal solution is found.


1 Answer 1


Yes, this analysis is essentially correct, although you are conflating branching and processing a node. Branching creates two children but does not necessarily process them right away. Processing means solving the LP relaxation at that node.

You are processing the nodes as a breadth first search (BFS). The paper mentions DFS, which is depth first search, but that must be a typo because the global bounds instead match your BFS.


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