Global Sensitivity Analysis via a Statistical Tolerance Approach

Area of Science:

• Optimization Theory
• Mathematical Modeling
• Computational Mathematics

Background:

• Sensitivity analysis and multiparametric programming are crucial for understanding optimization model behavior under parameter uncertainty.
• Existing methods often struggle with simultaneous variations in multiple input parameters, especially when these parameters are described by multivariate probability distributions.

Purpose of the Study:

• To develop a robust framework for sensitivity analysis in optimization when objective and constraint parameters (RIM parameters) vary jointly.
• To introduce a tolerance approach using principal component analysis for defining confidence sets for random input parameters.
• To extend the tolerance approach to handle cases with multiple optimal bases within the tolerance region.

Main Methods:

• Introduction of a tolerance approach based on principal component analysis to define distribution-suited tolerance regions.
• Extension of the tolerance approach to address multiple optimal bases by studying critical regions.
• Development of a computational algorithm to identify critical regions within the RIM parameter space that cover a given tolerance region.

Main Results:

• A novel tolerance approach is proposed for analyzing simultaneous variations in optimization model inputs.
• Theoretical insights into the geometric properties of critical regions are provided, enhancing the understanding of joint parameter variations.
• A computational algorithm is presented for finding critical regions relevant to sensitivity analysis.

Conclusions:

• The proposed framework offers a deeper geometric understanding of critical regions in parametric programming with jointly varying parameters.
• The developed methods are evaluated through experiments in sensitivity analysis, inventory management model predictive control, and large-scale optimization problems.
• This work advances the theory and computational methods for sensitivity analysis in optimization under multivariate parameter uncertainty.