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

which find good solutions but can not be proven to be optimal. Exact solutions can
typically be implemented successfully only for small problems, while heuristic methods
are used for larger problems where computation time of an exact method would become
prohibitive.
A commonly used exact algorithm is branch-and-bound, which branches the
original problem into successively smaller sub-problems. Each subset contains a relaxed
version of the original problem, which is easier to solve. The procedure continues until
each branch has resulted in either a feasible solution or is shown to contain no solution
better than one already obtained. Branch-and-bound methods result in locating the global
optimum66.
Similarly, dynamic programming also takes advantage of problem
decomposition, where the optimal solution to a given problem is expressed in terms of
optimal solutions of smaller sub-problems. 67
One of the most commonly known heuristic algorithms is the nearest neighbor
algorithm, also referred to as the greedy algorithm. In this method, the local optimum is
chosen at each step. For the classic TSP, for example, this would equate to choosing the
closest city at each step, until all of the cities have been visited. Another common
heuristic method is to use minimum spanning trees. A spanning tree is a collection of (n-
1) edges which join all n cities into a tree-structure. This can then be extrapolated to
create a tour, where each city is only visited once. While the heuristic methods do not
solve for the optimum solution, they can at least provide lower and upper bounds on the
optimum. However, one of the biggest challenges of heuristic methods is establishing
performance guarantees – i.e., bounds on how far the solution will be from the optimum
in the worst case66.
The classic TSP has many analogous features to the asteroid rendezvous problem,
where the “distance” between each asteroid is instead a combination of propellant
consumption and time of flight. Some major differences do exist, however, between the
classic TSP and the asteroid rendezvous problem. First, the asteroid problem does not
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