Representing Sampled Pareto Frontiers as Parameterized Continuous Manifolds Using Self-Organizing Maps.

This paper describes a method for defining a coordinate system to parameterize a sampled Pareto frontier of a continuous multi-attribute design problem using a modified self-organizing map. By defining such a coordinate system, the design problem may be reformulated from y = f (x) to (y,x) = g(ψ), where x is a vector of design variables, y is a vector of attributes, and ψ is a vector of barycentric coordinates. Exploration of the design problem using yr as the independent variables has the following desirable properties: 1) Every vector yr corresponds to a Pareto efficient design, and every Pareto efficient design has a corresponding ψ. 2) The number of ψ coordinates is equal to the number of attributes, regardless of the number of design variables. 3) Each attribute yi has a corresponding coordinate ψ, such that the sign of ∂yi/∂ψi is positive if the objective is to maximize y, and negative if the objective is to minimize yi (i.e., the attributes improve monotonically as their corresponding coordinates increase). This approach, named the "Pareto simplex self-organizing map" is easily implemented as a postprocessing step to sampling a Pareto frontier with multi-objective optimization. The construction of the coordinates and their use in design space exploration is demonstrated on a conceptual wing design problem. [ABSTRACT FROM AUTHOR]