Transforming Design Spaces Using Pareto-Laplace Filters

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.

Figures (3)

• Figure 1: Schematic representation of design landscape integral transform for a single design objective. (Top middle) A design problem exhibits a landscape of potential solutions distributed over a solution space $\mathcal{S}$ in coordinates $(\mathcal{O},x_\perp)$ according to the level at which they satisfy the design objective $\mathcal{O}$ and a set of other design variables $x_\perp$ (represented here schematically as a single variable). There may be $\Omega_\perp$ solutions that satisfy the design objective at some fixed level $\mathcal{O}$. We propose to filter the solution space via the application of a integral transform, where $Z$ computes the volume of the solution space depending a parameter $\beta$ that controls the degree of filtering of poor design solutions (which we take as large $\mathcal{O}$). For $\beta=0$ (lower left) there is no filtering, however increasing $\beta$ (lower right) effectively "pinches" the landscape for large $\mathcal{O}$. Increasingly large values of $\beta$ leave effectively larger relative contributions from near optimal solutions, with only the optimal solution remaining in the limit $\beta\to\infty$.
• Figure 2: The Pareto-Laplace filtering framework can be interpreted geometrically, statistically, and physically. This framework provides the means to operationalize multiple different forms of design investigation in terms of one or more of the three perspectives.
• Figure 3: Schematic representation of Pareto-Laplace transform for a situation with multiple minima. In the pre-filtered picture (top panels), the sub-leading, local minimum is at a lower value of $\mathcal{O}$ in the scenario depicted in the left image compared to the scenario in the right image. In the post-filtered picture (lower panels) the filter more strongly "pinches" the region of the solution space near the local minimum with larger $\mathcal{O}$ (right) compared to the one with lower $\mathcal{O}$ (left), illustrating the essence of the effect anticipated from Eq. \ref{['eq:Zmultiple']}.