Transforming Design Spaces Using Pareto-Laplace Filters

TL;DR

This paper introduces the Pareto-Laplace transform framework, a novel approach that enhances understanding of complex optimization problems by providing new geometric, statistical, and physical insights, unifying existing methods.

Contribution

The paper presents a new Pareto-Laplace integral transform framework that generalizes existing approaches and offers deeper insights into optimization problems across various representations.

Findings

Framework admits geometric, statistical, and physical representations

Unifies known approaches as special cases

Enables new insights into relationships between objectives and outcomes

Abstract

Optimization is a critical tool for addressing a broad range of human and technical problems. However, the paradox of advanced optimization techniques is that they have maximum utility for problems in which the relationship between the structure of the problem and the ultimate solution is the most obscure. The existence of solution with limited insight contrasts with techniques that have been developed for a broad range of engineering problems where integral transform techniques yield solutions and insight in tandem. Here, we present a ``Pareto-Laplace'' integral transform framework that can be applied to problems typically studied via optimization. We show that the framework admits related geometric, statistical, and physical representations that provide new forms of insight into relationships between objectives and outcomes. We argue that some known approaches are special cases of this framework, and point to a broad range of problems for further application.