(558b) A Unifying Abstraction for Infinite-Dimensional Optimization

Infinite-dimensional optimization problems contain variables that live in infinite-dimensional spaces such as states over a space-time field or states under uncertainty [1]. This class of problems commonly embeds complex modeling elements that include: measures (e.g., multi-dimensional integrals), differential algebraic equations (DAEs), and partial differential equations (PDEs). These elements, in combination with the presence of infinite-dimensional decision variables, make this problem class challenging to model and solve, since most optimization algorithms are intended for finite-dimensional spaces (e.g., linear and nonlinear programming) [2].

Infinite-dimensional problems encompass a wide breadth of optimization fields that include stochastic optimization [3], dynamic optimization [4], PDE-constrained optimization [5], and combinations (e.g., stochastic PDEs) [6]. For instance, decision variables in stochastic problems are random variables that are dependent on infinite-dimensional random spaces with objectives and constraints that employ stochastic measure operators over those spaces (e.g., risk measures and chance constraints) [3]. Applications pertaining to these optimization fields include model predictive control [7], process design [8], parameter estimation [9], reliability analysis [10], design of dynamic experiments [11], and more.

Typically, optimization problems of this class are solved by transforming them into finite-dimensional representations via discretization (e.g., finite differences) [12]. This "transcription" process enables the use of standard optimization solution approaches, and is so predominant as a modeling/transformation paradigm that application classes often forego infinite-dimensional representations and the use of other transformation techniques (e.g., projection unto orthogonal basis functions [1]). Moreover, although problems in the aforementioned optimization fields are often identified as infinite-dimensional optimization problems, there exists a conceptual gap in abstracting these problems rigorously through a common lens. Coherent abstractions play a key role in facilitating innovative advancements and enabling general modeling languages [13].

To address this conceptual gap, we present a unifying abstraction for characterizing and modeling infinite-dimensional optimization problems. Specifically, we propose a measure-centric modeling paradigm to capture a wide breadth of optimization fields and promote novel theoretical insights/techniques. Our abstraction forms the basis for a Julia-based, infinite-dimensional optimization modeling implementation entitled InfiniteOpt.jl [14]. We demonstrate how characterizing problems in this abstraction readily encourages new theoretical crossover and novel problem formulations, and even engenders new optimization fields (e.g., random field optimization and optimization using basis functions). We also discuss how this perspective decouples traditional finite transformation techniques across the subsequent disciplines and promotes the use of non-traditional solution methodologies.

References:

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[14] Joshua L Pulsipher, Weiqi Zhang, and Victor M Zavala. InfiniteOpt.jl -- A Julia package for modeling infinite-dimensional optimization problems. Zenodo, 10.5281/zenodo.4291106, 2020.