design space pruning heuristics and global optimization method for ...

As aforementioned, branch-and-bound is one type of algorithm that can be
applied to the traveling salesman problem. It can also be applied to the more general
integer programming problem, 65,73 which was described in Equation 12. The idea behind
this algorithm is to solve versions of the relaxed problem, which removes the integer
restriction from the variables, in order to set bounds on the integer problem. These
bounds are then used to eliminate branches of the branch-and-bound tree until all nodes
are either pruned or solved. This yields the optimal integer solution of the original
integer programming problem. Figure 8 illustrates a segment of a branch-and-bound
search tree.
S
x 1
=0 x 1
=1
S 0 S 1
x 2
=0 x 2
=1
S 10 S 11 S 111
x 3
=0
x 3
=1
S 110
Figure 8: Portion of a branch-and-bound search tree.
In describing the general branch-and-bound algorithm, L is a collection of integer
i
i i
programs, each of which takes the form z max{ cx : x ∈S
};
S ⊆ S
IP
= . Associated with
each problem in L is an upper bound
i
z ≥
z
i
IP
. First, the optimal relaxed solution is
determined for the problem as a whole. If the variables that yield the optimal solution all
take integer values, then the algorithm is terminated and the optimal solution is found.
Generally, this is not the case, so branches of the tree must begin to be enumerated.
There are numerous techniques for deciding the order in which these should be evaluated,
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