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Risk-Averse and Distributionally Robust Optimization: Methodology and Applications
Rahimian, Hamed

2018, Doctor of Philosophy, Ohio State University, Industrial and Systems Engineering.
Many decision-making problems arising in science, engineering, and business involve
uncertainties. One way to address these problems is to use stochastic optimization.
A crucial task when building stochastic optimization models is quantifying a
probability distribution to represent the uncertainty. Most often, partial information
about the uncertainty is available through a series of historical data. In such circumstances,
classical stochastic optimization models rely on approximating the underlying
probability distribution. However, in many real-world applications, the underlying
probability distribution cannot be accurately determined, even when historical data
are available. This distributional ambiguity might lead to highly suboptimal decisions.
An alternative approach to handle such an issue is to use distributionally robust
stochastic optimization
(DRSO for short), which assumes the underlying probability
distribution is unknown but lies in an ambiguity set of distributions.

Many existing studies on DRSO focus on how to construct the ambiguity set and
how to transform the resulting DRSO into equivalent (well-studied) models such as
mixed-integer programming and semide finite programming. This dissertation, however,
addresses more fundamental questions, in a different manner than the literature.
An overarching question that motivates most of this dissertation is which
scenarios/uncertainties are critical to a stochastic optimization problem? A major
contribution of this dissertation is a precise mathematical defi nition of what is meant by
a critical scenario and investigation on how to identify them for DRSO. As has
never been done before for DRSO (to the best of our knowledge), we introduce the
notion of effective and ineffective scenarios for DRSO.

This dissertation considers DRSOs for which the ambiguity set contains all probability
distributions that are not far---in the sense of the so-called total variation
distance
---from a nominal distribution (which may be obtained from data). This dissertation
then identifi es effective scenarios for two classes of DRSO problems formed
via the total variation distance: (1) a class of convex stochastic optimization problems
with a discrete sample space and (2) a class of inventory problems with a continuous
sample space.

All these classes of DRSO problems have equivalent risk-averse optimization problems
that lay the foundation to identify effective scenarios. We elaborate how effective
scenarios, along with other notions, can be used to choose an appropriate size for
the ambiguity set of distributions. Then, we devise customized algorithms to solve
DRSO formed via the total variation distance. Moreover, we survey existing algorithms
to solve a closely related risk-averse optimization problem to those induced by
the studied DRSO problems, and we propose new variations. Finally, to highlight the
practical relevance of our findings, we implement all our modeling, theoretical, and
computational results to solve problems arising in environment, energy, healthcare,
and finance.
Guzin Bayraksan, PhD (Advisor)
Antonio Conejo, PhD (Committee Member)
David Sivakoff, PhD (Committee Member)
242 p.

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Rahimian, H. (2018). Risk-Averse and Distributionally Robust Optimization: Methodology and Applications. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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Rahimian, Hamed. "Risk-Averse and Distributionally Robust Optimization: Methodology and Applications." Electronic Thesis or Dissertation. Ohio State University, 2018. OhioLINK Electronic Theses and Dissertations Center. 23 Oct 2018.

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Rahimian, Hamed "Risk-Averse and Distributionally Robust Optimization: Methodology and Applications." Electronic Thesis or Dissertation. Ohio State University, 2018. https://etd.ohiolink.edu/

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