Bezier Simplex Robustness Evaluation for Pareto Fronts
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Solution Overview
Problem
In multi-objective optimization problems, especially those with a large number of objective functions, the curse of dimensionality leads to sparse data distribution, making it difficult to determine the robustness of solutions against noise when visualizing the Pareto front.
Innovation Solution
A Bezier simplex is fitted to the solutions over the Pareto front, generating data points, calculating gradients at these points, and outputting evaluation information on the robustness of each solution based on the gradient, facilitating the determination of robust solutions.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If a large number of objective functions are used in multi-objective optimization, then the comprehensiveness of the optimization result is improved, but the data distribution becomes sparse due to the curse of dimensionality
Solution Approach 1:
The patent introduces a Bezier simplex as an intermediary mathematical model to represent the Pareto front. This mediator allows the system to work with a compact set of control points rather than sparse high-dimensional data, enabling effective visualization and robustness evaluation even when the original optimization data is sparse due to many objective functions.
2Loss of information
If the Pareto front is visualized in high-dimensional space, then the information about multiple solutions is preserved, but it becomes difficult to determine the robustness of solutions against noise
Solution Approach 1:
The patent transforms the high-dimensional Pareto front data into a lower-dimensional representation using a Bezier simplex model. By mapping solutions onto a parametric surface defined by control points, the system enables 2D or 3D visualization that preserves essential information while making robustness evaluation through gradient calculation feasible and intuitive.
Solution Approach 2:
The patent replaces direct visual inspection of high-dimensional sparse data with a mathematical modeling approach using Bezier surfaces. The gradient of the Bezier simplex serves as a quantitative metric for robustness, substituting subjective visual assessment with objective mathematical measurement that can be computed and visualized effectively.
Data Source
AI summary
A non-transitory computer-readable recording medium stores an output program for causing a computer to execute a process including: acquiring a plurality of solutions over a Pareto front; fitting a Bezier simplex to the plurality of solutions that has been acquired; calculating a gradient of the Bezier simplex at a plurality of data points over the Bezier simplex that has been fit; and for each of the plurality of data points that has been generated, outputting evaluation information on robustness of a solution corresponding to the data point based on the gradient that has been calculated.