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Probable Event Constrained Optimization and A Data-embedded Solution Paradigm

Qifeng Li

TL;DR

This work introduces Probable Event Constrained Optimization (PECO), a novel decision-making framework that guarantees feasibility for all realizations of uncertain parameters whose joint probability exceeds a user-defined threshold $\alpha$. PECO replaces traditional chance constraints with Probable Event Constraints (PEC), and to solve PECO in nonlinear and nonconvex settings, the authors propose a data-embedded deterministication (DeDA) that substitutes uncertainty with historical data, bypassing the need for explicit probability models. A key advance is the Strategic Data Selection (SDS) method, which dramatically reduces the embedded data required while maintaining solution quality. The PECO framework is demonstrated on optimal power flow under uncertainty, where DeDA and SDS yield accurate solutions with substantial computational savings, illustrating PECO’s practical impact for engineering systems facing uncertain but frequently observed conditions.

Abstract

This paper solves a new class of optimization problems under uncertainty, called Probable Event Constrained Optimization (PECO), which optimizes an objective function of decision variables and subjects to a set of Probable Event Constraints (PEC). This new type of constraint guarantees that optimal solutions are feasible for all uncertain events whose joint probabilities are greater than a user-defined threshold. The PEC can be used as an alternative to the conventional chance constraint, while the latter cannot guarantee the solution's feasibility to high-probability uncertain events. Given that the existing solution methods of optimization problems under uncertainty are not suitable for solving PECO problems, we develop a novel data-embedded solution paradigm that uses historical measurements/data of the uncertain parameters as input samples. This solution paradigm is conceptually simple and allows us to develop effective data-reduction schemes which reduce computational burden while preserving high accuracy.

Probable Event Constrained Optimization and A Data-embedded Solution Paradigm

TL;DR

This work introduces Probable Event Constrained Optimization (PECO), a novel decision-making framework that guarantees feasibility for all realizations of uncertain parameters whose joint probability exceeds a user-defined threshold . PECO replaces traditional chance constraints with Probable Event Constraints (PEC), and to solve PECO in nonlinear and nonconvex settings, the authors propose a data-embedded deterministication (DeDA) that substitutes uncertainty with historical data, bypassing the need for explicit probability models. A key advance is the Strategic Data Selection (SDS) method, which dramatically reduces the embedded data required while maintaining solution quality. The PECO framework is demonstrated on optimal power flow under uncertainty, where DeDA and SDS yield accurate solutions with substantial computational savings, illustrating PECO’s practical impact for engineering systems facing uncertain but frequently observed conditions.

Abstract

This paper solves a new class of optimization problems under uncertainty, called Probable Event Constrained Optimization (PECO), which optimizes an objective function of decision variables and subjects to a set of Probable Event Constraints (PEC). This new type of constraint guarantees that optimal solutions are feasible for all uncertain events whose joint probabilities are greater than a user-defined threshold. The PEC can be used as an alternative to the conventional chance constraint, while the latter cannot guarantee the solution's feasibility to high-probability uncertain events. Given that the existing solution methods of optimization problems under uncertainty are not suitable for solving PECO problems, we develop a novel data-embedded solution paradigm that uses historical measurements/data of the uncertain parameters as input samples. This solution paradigm is conceptually simple and allows us to develop effective data-reduction schemes which reduce computational burden while preserving high accuracy.
Paper Structure (42 sections, 15 theorems, 51 equations, 4 figures, 3 tables, 2 algorithms)

This paper contains 42 sections, 15 theorems, 51 equations, 4 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. The proposed method
  4. Related work
  5. Contributions
  6. The Probable Event Constraint (PEC)
  7. Illustrative examples of Definition \ref{['def:srd']}
  8. Relations between PEC and the classic chance constraint
  9. Applying the scenario method to solve PECO
  10. The Proposed Solution Paradigm for PECO: Data-embedded Deterministication
  11. DeDA($\mathcal{D}_{\alpha}^z$): A data-embedded deterministic approximation for PECO
  12. The Accuracy of Approximating PECO with DeDA($\mathcal{D}^z_{\alpha}$)
  13. Obtaining $\mathcal{D}_\alpha$: Defining the joint probability of a scenario based on maximum likelihood and optimal bandwidth using historical data
  14. Discussions and Summary
  15. A proper estimation of $\bar{B}_{\forall,\alpha}^{\rm OS}$
  16. ...and 27 more sections

Key Result

Proposition 2.1

\newlabelpro:feasisetpccsp0 $\mathcal{X}_{\rm P}(\alpha_1) \supseteq \mathcal{X}_{\rm P}(\alpha_2)$ if $\alpha_1 \ge \alpha_2$.

Figures (4)

  • Figure 1: An illustrative example of Definition \ref{['def:srd']}.
  • Figure 1: A pictorial interpretation of boundary-forming, active, and inactive data points in a 2-D $x$-space, where constraints 1-4 are the boundary-forming constraints of the feasible space, 3 and 4 are the boundary-forming constraints of the optimal solution, 3-5 are active constraints, and the rest are inactive constraints. A boundary-forming/active data point is a data point that contributes at least one boundary-forming/active constraint.
  • Figure 2: An illustrative example of $\Upxi_\alpha$.
  • Figure 2: A pictorial interpretation of $\zeta$.

Theorems & Definitions (23)

  • Definition 1.1: on a probable event
  • Definition 1.2: on key terminologies
  • Proposition 2.1
  • Proposition 2.2: on the relations between $\mathcal{X}_{\rm P}$ and $\mathcal{X}_{\rm C}$
  • Proposition 2.3
  • Proposition 3.1
  • Definition 3.2: on a deterministic equivalent of PECO
  • Definition 3.3: on the boundary-forming data points of the feasible space of DeDA
  • Proposition 3.4: on the relations between the feasible spaces of DeDA($\mathcal{D}_\alpha^z$) and PECO
  • Definition 3.5: on the boundary-forming data points of an optimal solution
  • ...and 13 more