Probability Distribution-free General Scenario Programming

TL;DR

This paper introduces a distribution-free approach for general optimization under uncertainty, utilizing historical data for deterministic equivalents and innovative scenario reduction techniques, applicable to both exogenous and endogenous uncertainties.

Contribution

It proposes a novel PD-free deterministic equivalent construction applicable to general optimization problems and develops methods for effective scenario reduction.

Findings

The PD-free method generalizes beyond chance-constrained optimization.

Active and repeated scenario properties are characterized.

Scenario reduction methods significantly decrease required scenarios.

Abstract

This paper presents a novel solution paradigm of general optimization under both exogenous and endogenous uncertainties. This solution paradigm consists of a probability distribution (PD)-free method of obtaining deterministic equivalents and an innovative approach of scenario reduction. First, dislike the existing methods that use scenarios sampled from pre-known PD functions, the PD-free method uses historical measurements of uncertain variables as input to convert the logical models into a type of deterministic equivalents called General Scenario Program (GSP). Our contributions to the PD-free deterministic equivalent construction reside in generalization (making it applicable to general optimization under uncertainty rather than just chance-constrained optimization) and extension (enabling it to the problems under endogenous uncertainty via developing an iterative and a non-iterative frameworks). Second, this paper reveals some unknown properties of the PD-free deterministic equivalent construction, such as the characteristics of active scenarios and repeated scenarios. Base on this discoveries, we propose a concept and methods of strategic scenario selection which can effectively reduce the required number of scenarios as demonstrated in both mathematical analysis and numerical experiments. Numerical experiments are conducted on two typical smart grid optimization problems under exogenous and endogenous uncertainties.