A Unifying Modeling Abstraction for Infinite-Dimensional Optimization
A Unifying Modeling Abstraction for Infinite-Dimensional Optimization
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DOI：
10.1016/j.compchemeng.2021.107567
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发表时间：
2021-06
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影响因子：
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通讯作者：
J. Pulsipher;Weiqi Zhang;Tyler J. Hongisto;V. Zavala
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作者：
J. Pulsipher;Weiqi Zhang;Tyler J. Hongisto;V. Zavala
无限维优化（InfiniteOpt）问题涉及建模组件（变量、目标和约束），这些组件是在无限维域上定义的函数。例子包括连续时间动态优化（时间是无限域，分量是时间的函数），PDE优化问题（空间和时间是无限域，分量是时空的函数），以及随机和半无限优化（随机空间是无限域，分量是这样的随机空间的函数）。InfiniteOpt问题也由这些问题类的组合引起（例如，随机PDE优化）。考虑到目标和约束的无限维性质，人们通常需要定义适当的量（度量）来正确地提出问题。此外，InfiniteOpt问题通常需要转换为有限维表示，以便可以数值求解。在这项工作中，我们提出了一个统一的抽象，便于建模，分析和解决InfiniteOpt问题。建议的抽象，使无限维域的一般治疗，并提供了一个以测量为中心的范例来处理相关的变量，目标和约束。这种抽象使我们能够跨学科转移技术，并以此识别新的、有趣的和有用的建模范例（例如，在时间域上定义的事件约束和风险度量）。我们的抽象作为一个直观的基于Julia的建模包的主干，我们称之为InfiniteOpt.jl。我们展示了不同的工程案例研究中出现的发展。
Infinite-dimensional optimization (InfiniteOpt) problems involve modeling components (variables, objectives, and constraints) that are functions defined over infinite-dimensional domains. Examples include continuous-time dynamic optimization (time is an infinite domain and components are functions of time), PDE optimization problems (space and time are infinite domains and components are functions of space-time), as well as stochastic and semi-infinite optimization (random space is an infinite domain and components are a function of such random space). InfiniteOpt problems also arise from combinations of these problem classes (e.g., stochastic PDE optimization). Given the infinite-dimensional nature of objectives and constraints, one often needs to define appropriate quantities (measures) to properly pose the problem. Moreover, InfiniteOpt problems often need to be transformed into a finite dimensional representation so that they can be solved numerically. In this work, we present a unifying abstraction that facilitates the modeling, analysis, and solution of InfiniteOpt problems. The proposed abstraction enables a general treatment of infinite-dimensional domains and provides a measure-centric paradigm to handle associated variables, objectives, and constraints. This abstraction allows us to transfer techniques across disciplines and with this identify new, interesting, and useful modeling paradigms (e.g., event constraints and risk measures defined over time domains). Our abstraction serves as the backbone of an intuitiveJulia-based modeling package that we callInfiniteOpt.jl. We demonstrate the developments using diverse case studies arising in engineering.