This dissertation presents unifying modeling abstractions for characterizing, analyzing, and solving infinite-dimensional optimization (InfiniteOpt) problems that are motivated by emergent applications in engineering. InfiniteOpt problems involve modeling components (variables, objectives, and constraints) that are functions defined over infinite domains. Examples include continuous-time dynamic optimization (time is an infinite domain and components are a function of time), PDE optimization problems (space and time are infinite domains and components are a function 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). Applications include model predictive control, process design, parameter estimation, reliability analysis, flexibility analysis, design of dynamic experiments, and more.

InfiniteOpt problems are often solved via discretization (e.g., finite differences), this is so predominant that application classes often forego InfiniteOpt representations and the use of other transformation techniques (e.g., projection unto orthogonal basis functions). Moreover, there exists a conceptual gap in abstracting these diverse optimization disciplines rigorously through a common lens. Coherent abstractions play a key role in facilitating innovative advancements and enabling general modeling languages.

To address this conceptual gap, we present a unifying abstraction for characterizing and modeling InfiniteOpt problems. This abstraction allows us to transfer techniques across disciplines and enable new modeling paradigms that include generalized risk measures, random field optimization, event-constrained optimization, and lifting-function based parameter estimation. Our abstraction serves as the backbone of an intuitive Juliabased modeling package called InfiniteOpt.jl which enables cutting-edge research and makes these advanced modeling techniques accessible.

Finally, we apply these principles to develop advanced abstractions for measuring and designing the flexibility and reliability of complex process systems. Flexibility denotes the ability of a system to maintain feasible operation in response to random fluctuations (i.e., continuous disturbances), and reliability assesses the ability of a system to continue feasible operation when subjected to random component failure (i.e., discrete events). Our proposed approaches facilitate the measuring and design of system flexibility/reliability that are more tractable and accurate relative to existing techniques.