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