(119h) Advanced Solution Techniques for Event Constrained Programming

Infinite-dimensional optimization (InfiniteOpt) problems encompass a wide range of challenging disciplines such as stochastic, dynamic, and PDE-constrained optimization where variables and constraints are indexed over continuous domains (i.e., variables are functions/manifolds) [1]. Here, applications include model predictive control [2], dynamic parameter estimation [3], portfolio planning [4], and optimal design [5]. Within these problem classes, it is common to relax constraints over a portion of the domain (e.g., chance constraints and soft constraints) to enhance feasibility and/or model application-specific considerations. Soft constraints from optimal control typically employ weighted penalty terms or slack variables in the objective to achieve this [6]; however, selecting weighting parameters such that the constraints are enforced over a certain portion of the domain is nontrivial. In stochastic optimization, chance constraints enforce constraints to a certain probability relative to a collection of possible random scenarios [7], but these use intersection logic (i.e., and operators) to aggregate constraints which makes them conservative relative to the complex logic that prevails in many applications.

We have developed event constraints as a new constraint class for general InfiniteOpt problems that enable us to pose relaxed constraints using intuitive parameters and arbitrary constraint aggregation logic. Event constraints generalize chance constraints for general InfiniteOpt domains (e.g., deterministic dynamic formulations) and complex constraint logic. These complex objects provide rich modeling capabilities but can be difficult to solve/reformulate due to the complexity in handling arbitrary logic and the scalability limitations that come with using binary variables (e.g., those stemming from big-M reformulations) in nonconvex formulations [8].

To address these challenges, we propose several approaches to efficiently solve event constrained formulations and straightforwardly handle arbitrary constraint logic. First, we present a generalized disjunctive programming (GDP) formulation which enables us to model complex constraint logic under the intuitive abstraction provided by established GDP tools [9]. Moreover, this allows us to utilize diverse solution approaches such as convex hull, strengthened big-M, and logic-based outer approximation that leverage the unique structure of GDP formulations [10]. We also propose a class of continuous approximations for event constraints by extending approaches from the chance constraint literature such as conditional-value-at-risk (CVaR) approximation and sigmoidal approximation [11, 12]. These negate the need for binary variables and can achieve tight approximations. We illustrate these findings using diverse case studies in stochastic, dynamic, and PDE-constrained optimization.

References

[1] Joshua L Pulsipher, Weiqi Zhang, Tyler J Hongisto, and Victor M Zavala. “A unifying modeling abstraction for infinite-dimensional optimization.” Computers & Chemical Engineering, 156:107567, 2022.

[2] James Blake Rawlings, David Q Mayne, and Moritz Diehl. “Model predictive control: theory, computation, and design,” volume 2. Nob Hill Publishing Madison, WI, 2017.

[3] Sungho Shin, Ophelia S Venturelli, and Victor M Zavala. “Scalable nonlinear programming framework for parameter estimation in dynamic biological system models.” PLoS computational biology, 15(3):e1006828, 2019.

[4] Dentcheva, Darinka, and Andrzej Ruszczyński. "Portfolio optimization with stochastic dominance constraints." Journal of Banking & Finance 30.2, 433-451, 2006.

[5] Amalia Nikolopoulou and Marianthi G Ierapetritou. “Optimal design of sustainable chemical processes and supply chains: A review.” Computers & Chemical Engineering, 44:94–103, 2012.

[6] Eric C Kerrigan and Jan M Maciejowski. “Soft constraints and exact penalty functions in model predictive control.” 2000.

[7] braham Charnes and William W Cooper. “Chance-constrained programming.” Management science, 6(1):73–79, 1959.

[8] Kyri Baker and Bridget Toomey. “Efficient relaxations for joint chance constrained ac optimal power flow.” Electric Power Systems Research, 148:230–236, 2017.

[9] Ignacio E Grossmann. “Advanced optimization for process systems engineering.” Cambridge University Press, 2021.

[10] Qi Chen, Emma S Johnson, David E Bernal, Romeo Valentin, Sunjeev Kale, Johnny Bates, John DSiirola, and Ignacio E Grossmann. “Pyomo.gdp: an ecosystem for logic based modeling and optimization development.” Optimization and Engineering, pages 1–36, 2021.

[11] Arkadi Nemirovski and Alexander Shapiro. “Convex approximations of chance constrained programs.” SIAM Journal on Optimization, 17(4):969–996, 2007.

[12] Yankai Cao and Victor M Zavala. “A sigmoidal approximation for chance-constrained nonlinear programs.” arXiv preprint arXiv:2004.02402, 2020.