(33b) Modeling Infinite-Dimensional Optimization Problems in Infiniteopt.Jl

Infinite-dimensional optimization problems are problems that contain variables that live in infinite-dimensional spaces (e.g., states over a space-time field or states under uncertainty) [1]. These types of problems also commonly embed measures (e.g., multi-dimensional integrals), differential algebraic equations (DAEs), and partial differential equations (PDEs). These features make these types of problems difficult to model and solve with existing modeling packages, which often can only handle finite-dimensional spaces [2].

Example problem classes of infinite-dimensional problems include stochastic programs, optimal control, optimization with PDEs, and combinations (e.g., stochastic PDEs and stochastic optimal control). For instance, decision variables in stochastic programs are random variables that are parameterized over infinite-dimensional spaces and objectives and constraints are expressed as measures over those spaces (e.g., expectations, chance constraints, and risk measures) [3]. Similarly, space-time models employ variables defined over continuous dimensions (i.e., time and/or spatial domains) [4-6]. Example applications include model predictive control and process design [7-9].

Typically, infinite-dimensional problems are transformed into finite-dimensional representations by using manual discretization (e.g., Euler, finite differences, Monte Carlo integration, and sparse grids) [10]. This "transcription" process is in general complex especially when involving multiple domains [2]. There are a number of domain-specific software implementations including Gekko, ACADO, and gPROMS that can handle certain problem forms (mostly dynamic), but are not capable of handling general dimensions [11-13]. Pyomo.dae provides more general capabilities that can handle temporal and spatial domains (e.g., via continuous sets) but handling of random variables is done independently (i.e., by nesting an infinite-dimensional problem within a stochastic programming problem).

In this talk, we present InfiniteOpt.jl, a modeling framework that can compactly express and automatically transform infinite-dimensional optimization problems. Our framework incorporates a unifying abstraction that can handle general continuous sets (e.g., intervals or random spaces), measures, and derivatives. InfiniteOpt.jl leverages Julia, an efficient programmatic environment that offers an intuitive symbolic API that is more accessible to users with limited background (similar to JuMP [14]). The framework also facilitates compact expressions of DAEs, PDEs, and other complex infinite-dimensional constraints. The framework also features a wide collection of automated transformation implementations that can readily be expanded by advanced users to enable specialized techniques. We provide diverse case studies to highlight these features.

Reference:

[1] Devolder, Olivier, François Glineur, and Yurii Nesterov. Solving infinite-dimensional optimization problems by polynomial approximation. Recent Advances in Optimization and its Applications in Engineering, pp. 31-40. Springer, Berlin, Heidelberg, 2010.

[2] Nicholson, Bethany, John D. Siirola, Jean-Paul Watson, Victor M. Zavala, and Lorenz T. Biegler. pyomo. dae: A modeling and automatic discretization framework for optimization with differential and algebraic equations. Mathematical Programming Computation 10, no. 2 (2018): 187-223.

[3] Birge, John R., and Francois Louveaux. Introduction to stochastic programming. Springer Science & Business Media, 2011.

[4] Zhou, Kemin, John Comstock Doyle, and Keith Glover. Robust and optimal control. Vol. 40. New Jersey: Prentice hall, 1996.

[5] Bertsekas, Dimitri P., Dimitri P. Bertsekas, Dimitri P. Bertsekas, and Dimitri P. Bertsekas. Dynamic programming and optimal control. Vol. 1, no. 2. Belmont, MA: Athena scientific, 1995.

[6] Hinze, Michael, René Pinnau, Michael Ulbrich, and Stefan Ulbrich. Optimization with PDE constraints. Vol. 23. Springer Science & Business Media, 2008.

[7] Rawlings, James B. Tutorial overview of model predictive control. IEEE control systems magazine 20, no. 3 (2000): 38-52.

[8] Stankiewicz, Andrzej I., and Jacob A. Moulijn. Process intensification: transforming chemical engineering. Chemical engineering progress 96, no. 1 (2000): 22-34.

[9] Straub, David A., and Ignacio E. Grossmann. Design optimization of stochastic flexibility. Computers & Chemical Engineering 17, no. 4 (1993): 339-354.

[10] Biegler, Lorenz T. An overview of simultaneous strategies for dynamic optimization. Chemical Engineering and Processing: Process Intensification 46, no. 11 (2007): 1043-1053.

[11] Beal, Logan DR, Daniel C. Hill, R. Abraham Martin, and John D. Hedengren. Gekko optimization suite. Processes 6, no. 8 (2018): 106.

[12] Houska, Boris, Hans Joachim Ferreau, and Moritz Diehl. ACADO toolkit—An open‐source framework for automatic control and dynamic optimization. Optimal Control Applications and Methods 32, no. 3 (2011): 298-312.

[13] Asteasuain, Mariano, Stella Maris Tonelli, Adriana Brandolin, and Jose Alberto Bandoni. Dynamic simulation and optimisation of tubular polymerisation reactors in gPROMS. Computers & Chemical Engineering 25, no. 4-6 (2001): 509-515.

[14] Dunning, Iain, Joey Huchette, and Miles Lubin. JuMP: A modeling language for mathematical optimization. SIAM Review 59, no. 2 (2017): 295-320.