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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1.2.3 Combinatorial Optimization<br />
Choosing the optimal asteroid sequence in an asteroid tour mission <strong>design</strong><br />
problem is by nature a combinatorial <strong>and</strong> integer <strong>optimization</strong> problem. The<br />
distinguishing feature of such problems is that the variables belong to a discrete set where<br />
there is not a continuum of alternatives. One can write the linear integer programming<br />
problem, which has no continuous variables, as follows 65 :<br />
{ cx : Ax b,x ∈ Z<br />
n }<br />
max ≤<br />
+<br />
(12)<br />
In Equation 12, Z n + is the set of nonnegative integral n-dimensional vectors, <strong>and</strong> x =<br />
(x 1 ,…,x n ) are the variables or unknowns. An instance of the problem is specified by the<br />
data (c, A, b), where c is an n-dimensional vector, A is an m x n matrix, <strong>and</strong> b is an m-<br />
dimensional vector. While this represents the general problem, it can be altered in a<br />
number of ways to represent myriad types of discrete <strong>optimization</strong> problems. One<br />
example of a <strong>for</strong>mulation of Equation 12 is the 0-1 knapsack problem, which is one of the<br />
most basic <strong>and</strong> common problems in combinatorial <strong>optimization</strong>. This problem deals<br />
with choosing a subset of projects to maximize the sum of their values while not<br />
exceeding some budget constraint. From Equation 12, the projects are represented by the<br />
variables x 1 ,…x n , where a value of 1 indicates that project j is in the subset <strong>and</strong> a value of<br />
0 indicates that it is not. The jth project has a cost of a j <strong>and</strong> a value of c j , <strong>and</strong> b represents<br />
the overall budget constraint.<br />
Another common type of combinatorial <strong>optimization</strong> problem, which is more<br />
directly applicable to the asteroid tour problem, is the traveling salesman problem. The<br />
classic version of this problem deals with solving the following scenario: given a set of<br />
cities <strong>and</strong> the distance between each pair of cities, determine the shortest route to visit<br />
each city exactly once while returning to the city of origin. 66<br />
Solution <strong>method</strong>s generally<br />
fall into two categories: algorithms <strong>for</strong> finding exact solutions <strong>and</strong> heuristic algorithms,<br />
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