A Comparative Theoretical and Computational Study on Robust Counterpart Optimization: I. Robust Linear Optimization...

Area of Science:

• Operations Research
• Optimization Theory
• Mathematical Programming

Background:

• Linear optimization and mixed integer linear optimization are crucial for complex decision-making.
• Handling uncertainty in optimization problems is a significant challenge.
• Existing robust optimization methods often rely on limited uncertainty set definitions.

Purpose of the Study:

• To investigate robust counterpart optimization techniques for linear and mixed integer linear optimization.
• To introduce and analyze novel uncertainty sets beyond those in existing literature.
• To derive robust counterpart formulations for various uncertainty scenarios and discuss their geometric relationships.

Main Methods:

• Development and analysis of various uncertainty sets: interval, adjustable box, ellipsoidal, polyhedral, and combinations thereof.
• Derivation of robust counterpart optimization formulations for left-hand side, right-hand side, and objective function uncertainties.
• Geometric analysis of the relationships between different uncertainty sets.

Main Results:

• Introduced and characterized new uncertainty sets, including adjustable box, pure ellipsoidal, pure polyhedral, and combined interval-ellipsoidal-polyhedral sets.
• Derived robust counterpart formulations for a comprehensive range of uncertainty sets and problem formulations.
• Numerical studies demonstrated the performance comparison of different robust counterpart optimization models.

Conclusions:

• The study provides a unified framework for analyzing various uncertainty sets in robust optimization.
• The proposed robust counterpart optimization techniques offer effective solutions for problems with uncertainty.
• Applications in refinery production planning and batch process scheduling highlight the practical utility of the developed methods.