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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draws upon the theory of niche <strong>and</strong> speciation in Darwinian evolution. 49<br />
The purpose of<br />
the sharing function is to degrade an individual’s fitness function based on its proximity<br />
to neighboring individuals. As a result, the largest number of individuals will converge<br />
to the local optimum with the best fitness value, with fewer individuals converging to<br />
optima with lesser fitness. The number of optima found by a genetic algorithm with<br />
sharing is a function of the size of the population. Another genetic algorithm variant<br />
addresses multi-objective problems. 50<br />
One way of dealing with multiple objectives is to<br />
solve <strong>for</strong> the Pareto-optimal set, which encompasses the set of non-dominated solutions.<br />
When comparing two solutions, x 1 <strong>and</strong> x 2 , x 1 dominates x 2 if (1) x 1 is no worse than x 2 in<br />
all objectives <strong>and</strong> (2) x 1 is strictly better than x 2 in at least one objective. There<strong>for</strong>e, the<br />
Pareto-optimal set contains all the solutions that are not dominated by any other<br />
solutions. This concept has been implemented in NSGA-II (non-dominated sorting<br />
genetic algorithm), developed by the Kanpur Genetic Algorithm Laboratory. 51<br />
Genetic algorithms have been applied to a number of different trajectory<br />
<strong>optimization</strong> problems, beginning with their application to ballistic (high thrust) transfers<br />
<strong>and</strong> gravity assist problems. 52,53,54,55<br />
For the high-thrust case, solving <strong>for</strong> a single<br />
trajectory is much less time-consuming <strong>and</strong> is generally done using a Lambert Solver.<br />
There<strong>for</strong>e, a genetic algorithm, even with the large number of required function calls, is<br />
appropriate <strong>for</strong> <strong>global</strong> <strong>optimization</strong>.<br />
Gage <strong>and</strong> Braun applied a genetic algorithm with a sharing function to impulsive<br />
Earth-Mars trajectories in order to optimize <strong>for</strong> launch date <strong>and</strong> time of flight54. Figure 4<br />
plots ∆V as a function of departure date <strong>and</strong> transfer time. As can be seen, this is a multimodal<br />
<strong>space</strong>, <strong>and</strong> <strong>for</strong> conceptual <strong>design</strong>, it is desirable to locate each of the local minima,<br />
which was successfully accomplished using a sharing function.<br />
20