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.

翻译：优化是解决人类与技术领域广泛问题的重要工具。然而，先进优化技术的悖论在于，它们对问题结构与最终解之间关系最为模糊的问题具有最大效用。通过有限洞察获取解的方法，与广泛应用于工程问题的积分变换技术形成对比——后者能同步获得解与洞察。本文提出一种"帕累托-拉普拉斯"积分变换框架，可应用于通常通过优化研究的问题。我们证明该框架具有几何、统计与物理表征，能为目标与结果间关系提供全新形式的洞察。我们论证某些已知方法属于该框架的特例，并指出该框架可进一步应用于更广泛的问题。