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

2.3.2 Branch-and-Bound
The branch-and-bound methodology considered here is a variation on the general
branch-and-bound method that uses linear programming relaxations, as presented in
Section 1.2.3. In an integer programming problem the relaxation step involves removing
the integer constraints and solving for the solution to the continuous problem. Relaxing
the integer constraints is not possible, however, when the integer variables represent
discrete asteroids choices. Therefore, in place of the linear programming relaxation, a
two-impulse Lambert solution is used to provide an upper bound on the low-thrust
solution for the branch-and-bound algorithm. The proposed branch-and-bound algorithm
is based on the conjecture that the two-impulse solutions provide a reliable upper-bound
to the low-thrust problem. This assumption will be examined on the sample problem.
S
ast 1 = 1 ast 1 = 2 ast 1 = 3
S 1 S 2 S 3
ast 2 = 4 ast 2 = 5 ast 2 = 6 ast 2 = 4 ast 2 = 5 ast 2 = 6 ast 2 = 4 ast 2 = 5 ast 2 = 6
S 14 S 15 S 16 S 24 S 25 S 26 S 34 S 35 S 36
ast 3 = 7 ast 3 = 7 ast 3 = 7 ast 3 = 7 ast 3 = 7 ast 3 = 7 ast 3 = 7 ast 3 = 7 ast 3 = 7
ast 3 = 8 ast 3 = 8 ast 3 = 8 ast 3 = 8 ast 3 = 8 ast 3 = 8 ast 3 = 8 ast 3 = 8 ast 3 = 8
S 147 S 148 S 157 S 158 S 167 S 168 S 247 S 248 S 257 S 258 S 267 S 268 S 347 S 348 S 357 S 358 S 367 S 368
Figure 13: Example branch-and-bound tree.
The search tree enumerates all possible asteroids sequences, a small example of
which is illustrated in Figure 13. The first branch represents the choice of the first
asteroid to visit from Earth. The next branch represents the second asteroid to visit in the
sequence and so on. The branch-and-bound tree is used only to solve for the optimal
asteroid sequence – the optimal departure date, flight times, and stay times must be
obtained using another method. In order to begin the algorithm, a known low-thrust
48