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Event Constrained Programming

Daniel Ovalle, Stefan Mazzadi, Carl D. Laird, Ignacio E. Grossmann, Joshua L. Pulsipher

TL;DR

A generalized disjunctive programming (GDP) representation of event constrained optimization problems is derived, which readily enables us to pose logical event conditions in a standard form and allows us to draw from a suite of GDP solution strategies that leverage the special structure of this problem class.

Abstract

In this paper, we present event constraints as a new modeling paradigm that generalizes joint chance constraints from stochastic optimization to (1) enforce a constraint on the probability of satisfying a set of constraints aggregated via application-specific logic (constituting an event) and (2) to be applied to general infinite-dimensional optimization (InfiniteOpt) problems (i.e., time, space, and/or uncertainty domains). This new constraint class offers significant modeling flexibility in posing InfiniteOpt constraints that are enforced over a certain portion of their domain (e.g., to a certain probability level), but can be challenging to reformulate/solve due to difficulties in representing arbitrary logical conditions and specifying a probabilistic measure on a collection of constraints. To address these challenges, we derive a generalized disjunctive programming (GDP) representation of event constrained optimization problems, which readily enables us to pose logical event conditions in a standard form and allows us to draw from a suite of GDP solution strategies that leverage the special structure of this problem class. We also extend several approximation techniques from the chance constraint literature to provide a means to reformulate certain event constraints without the use of binary variables. We illustrate these findings with case studies in stochastic optimal power flow, dynamic disease control, and optimal 2D diffusion.

Event Constrained Programming

TL;DR

A generalized disjunctive programming (GDP) representation of event constrained optimization problems is derived, which readily enables us to pose logical event conditions in a standard form and allows us to draw from a suite of GDP solution strategies that leverage the special structure of this problem class.

Abstract

In this paper, we present event constraints as a new modeling paradigm that generalizes joint chance constraints from stochastic optimization to (1) enforce a constraint on the probability of satisfying a set of constraints aggregated via application-specific logic (constituting an event) and (2) to be applied to general infinite-dimensional optimization (InfiniteOpt) problems (i.e., time, space, and/or uncertainty domains). This new constraint class offers significant modeling flexibility in posing InfiniteOpt constraints that are enforced over a certain portion of their domain (e.g., to a certain probability level), but can be challenging to reformulate/solve due to difficulties in representing arbitrary logical conditions and specifying a probabilistic measure on a collection of constraints. To address these challenges, we derive a generalized disjunctive programming (GDP) representation of event constrained optimization problems, which readily enables us to pose logical event conditions in a standard form and allows us to draw from a suite of GDP solution strategies that leverage the special structure of this problem class. We also extend several approximation techniques from the chance constraint literature to provide a means to reformulate certain event constraints without the use of binary variables. We illustrate these findings with case studies in stochastic optimal power flow, dynamic disease control, and optimal 2D diffusion.
Paper Structure (37 sections, 1 theorem, 81 equations, 22 figures, 14 tables, 1 algorithm)

This paper contains 37 sections, 1 theorem, 81 equations, 22 figures, 14 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Basic Notation and Background
  3. Chance Constraints
  4. Sample Average Approximation with big-M Constraints
  5. Individual Chance Constraint Reformulations
  6. Converting Joint Chance Constraints to Individual Chance Constraints
  7. Unifying Abstraction for InfiniteOpt Problems
  8. Generalized Disjunctive Programming
  9. Mathematical Programming with Complementarity Constraints
  10. Event Constraints
  11. General Formulation
  12. Generalizing the Domain
  13. Generalizing the Logic
  14. Previous Solution Strategies
  15. Event Constraint Solution Techniques
  16. ...and 22 more sections

Key Result

Proposition 1

Formulation eq:infinite_gdp is exact meaning that a Boolean variable $Y_i(d)= \text{True}$ if and only if constraints $h_i(d) \leq 0$ are satisfied for $i \in \mathcal{I}, \ d \in \mathcal{D}$. Consequently, the expectation in eq:gdp_event_constraint captures every realization where $h_i(d) \leq 0$

Figures (22)

  • Figure 1: A visual summary of how event constrained programming builds upon the theoretical foundation provided by chance constraints and how GDP can be used to model complex logic.
  • Figure 2: A depiction of how using infinite parameter, infinite variable, measure operator, and differential operator modeling objects allows us to capture formulations in stochastic, dynamic, and PDE-constrained optimization and combinations.
  • Figure 3: An illustration of the classical logical intersection at a particular value $\hat{d}$ of $d$ for a tertiary constraint system in \ref{['eq:chance_constr_analog']}. Using logical operators this is expressed $h_1(\hat{d}) \leq 0 \wedge h_2(\hat{d}) \leq 0 \wedge h_3(\hat{d}) \leq 0$.
  • Figure 4: Examples of generalized logic in a tertiary constraint system at a particular value $\hat{d}$ that encompasses more than just the logical intersection.
  • Figure 5: IEEE-14 power grid network topology.
  • ...and 17 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof