(47f) Event Constrained Optimization

Infinite-dimensional optimization (InfiniteOpt) problems feature constraints that are parameterized over infinite (indexing) domains (e.g., time, space, and/or uncertainty) due to their dependence on infinite variables (i.e., variables that are indexed over continuous domains) [1]. Commonly, these seek to enforce conditions/bounds on constraint values over the entire span of the infinite domain. This approach is particularly common in dynamic optimization (DO) and stochastic optimization (SO) [2, 3]. For instance, in DO, one may seek to keep time trajectories for controls/states below a certain threshold value for all times in a time horizon. In SO, one may seek to satisfy operational limits for all possible realizations of uncertainty (in this context the constraints are said to be enforced almost surely or with probability of one). These are also manifest by path constraints in PDE/dynamic optimization [4]. However, this InfiniteOpt constraint class can be overly restrictive for certain applications because each constraint must be satisfied over the entire span of the infinite domain. This limitation can be alleviated by enforcing InfiniteOpt constraints over a portion of the infinite domain and/or enforcing constraints on a measure of constraint functions (e.g., chance constraints) [1].

In SO, the above observations motivated the development of chance constraints which seek to enforce a stochastic constraint to a certain probability level (i.e., the constraint must be satisfied over a certain fraction of the uncertainty scenarios) [5]. Similarly, joint chance constraints enforce that a collection of constraints be satisfied jointly to a desired probability level [6]. Moreover, joint chance constraints can be interpreted as a generalization of other constraints since imposing a probability level of one is equivalent to enforcing the constraint collection almost surely [1]. Chance constraints have seen extensive use in a wide breadth of applications that include optimal power flow [7], model predictive control [8], flexibility analysis [9], portfolio optimization [10], reservoir operation [11], and process synthesis [12]. However, joint chance constraints can be overly restrictive since they enforce a condition jointly (i.e., on the intersection of) a constraint collection. Moreover, chance constraints have been limited to a SO context.

Through the lens of our unifying abstraction for InfiniteOpt problems [1], we can address these limitations by proposing a new generalized constraint class that we call event constraints. Event constraints consider the use of logical operators (e.g., AND, OR, NOT) to enforce a constraint on the probability of a particular event (encoded by the aggregation of InfiniteOpt constraints via logical operators). These produce classical joint chance constraints as a special case (a constraint collection is aggregated with AND operators) and can be used with general InfiniteOpt problems following our general treatment of probability measure operators in our unifying abstraction. Moreover, we propose efficient solution approaches that built upon propositional logic and general disjunctive programming [13]. Moreover, we motivate these new modeling constructs via illustrative case studies.

References:

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

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

[3] Powell, Warren B. "A unified framework for stochastic optimization." European Journal of Operational Research 275, no. 3 (2019): 795-821.

[4] Biegler, Lorenz T., Omar Ghattas, Matthias Heinkenschloss, and Bart van Bloemen Waanders. "Large-scale PDE-constrained optimization: an introduction." In Large-Scale PDE-Constrained Optimization, Springer, Berlin, Heidelberg, (2003): 3-13.

[5] Nemirovski, Arkadi, and Alexander Shapiro. "Scenario approximations of chance constraints." In Probabilistic and randomized methods for design under uncertainty, Springer, London, (2006): 3-47.

[6] Pagnoncelli, Bernardo K., Shabbir Ahmed, and Alexander Shapiro. "Sample average approximation method for chance constrained programming: theory and applications." Journal of optimization theory and applications 142, no. 2 (2009): 399-416.

[7] Lubin, Miles, Yury Dvorkin, and Line Roald. "Chance constraints for improving the security of ac optimal power flow." IEEE Transactions on Power Systems 34, no. 3 (2019): 1908-1917.

[8] Schwarm, Alexander T., and Michael Nikolaou. "Chance‐constrained model predictive control." AIChE Journal 45, no. 8 (1999): 1743-1752.

[9] Pulsipher, Joshua L., and Victor M. Zavala. "A scalable stochastic programming approach for the design of flexible systems." Computers & Chemical Engineering 128 (2019): 69-76.

[10] Pagnoncelli, Bernardo K., Shabbir Ahmed, and Alexander Shapiro. "Computational study of a chance constrained portfolio selection problem." Journal of Optimization Theory and Applications 142, no. 2 (2009): 399-416.

[11] Sreenivasan, K. R., and S. Vedula. "Reservoir operation for hydropower optimization: a chance-constrained approach." Sadhana 21, no. 4 (1996): 503-510.

[12] Esche, Erike, David Müller, Sebastian Werk, Ignacio E. Grossmann, and Günter Wozny. "Solution of chance-constrained mixed-integer nonlinear programming problems." In Computer Aided Chemical Engineering, vol. 38, Elsevier, (2016): 91-96.

[13] Grossmann, Ignacio E. Advanced optimization for process systems engineering. Cambridge University Press, (2021).