A Data-embedded Solution Paradigm for Nonconvex Probable Event Constrained Optimization

TL;DR

This paper proposes a novel data-embedded framework for solving nonconvex optimization problems under uncertainty, explicitly ensuring high-probability feasibility by leveraging historical data and machine learning techniques.

Contribution

It introduces Probable Event Constrained Optimization (PECO) and a data-embedded program (DEP) that directly incorporate data to handle nonlinear, nonconvex problems under uncertainty.

Findings

DEP effectively incorporates historical data for PECO.

The approach can handle nonlinear and nonconvex problems.

Machine learning aids in estimating data set sizes for solution robustness.

Abstract

This paper introduces a new modeling framework for optimization under uncertainty, called Probable Event Constrained Optimization (PECO). Unlike conventional chance-constrained formulations, which only limit the probability of constraint violation, PECO also explicitly requires feasibility for all events whose probability exceeds a prescribed threshold. This guarantees that solutions remain valid across all high-probability realizations of uncertainty. To solve PECO, we proposed a data-embedded program (DEP) which directly incorporates historical measurements of the uncertain parameters to obtain a deterministic approximation for PECO. While existing solution methods for optimization problems under uncertainty rely heavily on convexity or linearity assumptions, the proposed data-embedded solution paradigm provides a unique opportunity for solving nonlinear and nonconvex PECOs. The effectiveness of this approach depends on properly estimating the number of elements in the family of solution-determining data sets. As we enter the era of big data, this information can be properly estimated by leveraging the power of machine learning.