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# Multistage Stochastic Optimization Under Exogenous and Endogenous Uncertainty

• Author / Creator
Farough Motamed Nasab
• Stochastic optimization is field in mathematics that deals with decision making under uncertain conditions. Traditional methods for solving stochastic optimization problems such as scenario-tree methods usually result in large-size problems that suffer from the curse of dimensionality and require significant amount of time and computational resources. This study aims at investigating alternative methods referred to as decision-rule methods in order to reduce the problem size while preserving the solution optimality.

Uncertain parameters are generally classified into two categories: exogenous and endogenous
uncertainty. Endogenous uncertain parameters are revealed by decisions taken during the problem while exogenous uncertain parameters are independent of the problem decisions. This study investigates both types of uncertainty and employs linear decision rule to model the uncertainty. Two main solution methods are used in this study: partitioning and lifting. In partitioning method, the uncertainty set is partitioned into rectangular segments. At each partition, the binary variable is fixed, and the continuous variable is a linear combination of uncertain parameters. In lifting method, the uncertain parameter is lifted to a higher dimensional uncertainty set. The binary variable is formulated using 0-1 indicator functions that results in an adaptive binary variable. The continuous variable is a linear combination of lifted uncertain parameters that results in a piecewise linear solution. At each problem, after applying linear decision rule to the constraints, duality theorem for linear problems is used to convert the constraints to their robust deterministic counterpart.

A new framework based on lifting technique is developed in this study. The proposed framework can incorporate both adaptive binary and adaptive continuous variables for multistage stochastic problems. It results in adaptive discontinuous solution for continuous variables. The obtained results demonstrate that the developed framework is very flexible and results in significant computational and run time efficiency compared to traditional scenario-tree method.

• Subjects / Keywords
Spring 2021
• Type of Item
Thesis
• Degree
Doctor of Philosophy
• DOI
https://doi.org/10.7939/r3-78hf-rm97