Linear Many-Objective (NSGA-III)

Given a set of input-output pairs, the task is to optimize the parameters used across multiple linear functions of the form:

\[ y_1 = w_1 x_{11} + w_2 x_{12} + \cdots + w_n x_{1n} \]

\[ y_2 = w_1 x_{21} + w_2 x_{22} + \cdots + w_n x_{2n} \]

\[\cdots\]

\[ y_m = w_1 x_{m1} + w_2 x_{m2} + \cdots + w_n x_{mn} \]

where:

• \( x_{11}, x_{12}, \dots, x_{mn} \) are the input features.
• \( w_1, w_2, \dots, w_n \) are the unknown parameters (coefficients) of the linear functions that need to be optimized.
• \( y_1, y_2, \dots, y_m \) are the outputs corresponding to the given inputs for each linear function.

The goal is to optimize \( w_1, w_2, \dots, w_n \) using a multi-objective optimization approach so that the outputs are as close as possible to the objectives target outputs \( y_1, y_2, \dots, y_n \).

This problem uses the NSGA-III algorithm by default. NSGA-III keeps the solutions spread out across the objective space using a set of structured reference points, which scales better than NSGA-II once there are four or more objectives. You can still switch the parent selection to an NSGA-II variant from the parameters form if you prefer.