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[shell hinge]

@trim coral Specific three-objective Pareto optimization numerical experiment example, giving specific numerical values

[trim coral]

In a three-objective Pareto optimization numerical experiment, let's consider minimizing three objectives simultaneously: f1(x), f2(x), and f3(x), within a given feasible region. Suppose the feasible region is defined by the domain of x: [0, 1].

Let the three objective functions be:

1. f1(x) = x^3
2. f2(x) = (1 - x)^3
3. f3(x) = x^2 * (1 - x)

Now, we'll use numerical optimization techniques like the NSGA-II or MOEA/D algorithms to find the approximate set of Pareto optimal solutions for this multi-objective problem. After the optimization process, we may find a set of Pareto optimal solutions, for example:

1. x1 = 0.2, f1(x1) = 0.008, f2(x1) = 0.512, f3(x1) = 0.032
2. x2 = 0.5, f1(x2) = 0.125, f2(x2) = 0.125, f3(x2) = 0.125
3. x3 = 0.8, f1(x3) = 0.512, f2(x3) = 0.008, f3(x3) = 0.032

In this specific numerical example, these three solutions represent trade-offs between the three objectives, a Pareto frontier.

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[shell hinge]

@trim coral Give the matlab code of the NSGA-II algorithm