Dynamic decision problems affected by uncertain data are notoriously hard to solve due to the
presence of adaptive decision variables which must be modeled as functions or decision rules of
some (or all) of the uncertain parameters. All exact solution techniques suffer from the curse
of dimensionality while most solution schemes assume that the decision-maker cannot influence
the sequence in which the uncertain parameters are revealed.
The main objective of this thesis is to devise tractable approximation schemes for dynamic
decision-making under uncertainty. For this purpose, we develop new decision rule approximations
whereby the adaptive decisions are approximated by finite linear combinations of
prescribed basis functions.
In the first part of this thesis, we develop a tractable unifying framework for solving convex
multi-stage robust optimization problems with general nonlinear dependence on the uncertain
parameters. This is achieved by combining decision rule and constraint sampling approximations.
The synthesis of these two methodologies provides us with a versatile data-driven framework,
which circumvents the need for estimating the distribution of the uncertain parameters
and offers almost complete freedom in the choice of basis functions. We obtain a-priori probabilistic
guarantees on the feasibility properties of the optimal decision rule and demonstrate
asymptotic consistency of the approximation.
We then investigate the problem of hedging and pricing path-dependent electricity derivatives
such as swing options, which play a crucial risk management role in today’s deregulated energy
markets. Most of the literature on the topic assumes that a swing option can be assigned
a unique fair price. This assumption nevertheless fails to hold in real-world energy markets,
where the option admits a whole interval of prices consistent with those of traded instruments.
We formulate two large-scale robust optimization problems whose optimal values yield the endpoints
of this interval. We analyze and exploit the structure of the optimal decision rule to
formulate approximate problems that can be solved efficiently with the decision rule approach
discussed in the first part of the thesis.
Most of the literature on stochastic and robust optimization assumes that the sequence in which
the uncertain parameters unfold is independent of the decision-maker’s actions. Nevertheless,
in numerous real-world decision problems, the time of information discovery can be influenced by the decision-maker. In the last part of this thesis, we propose a decision rule-based approximation
scheme for multi-stage problems with decision-dependent information discovery.
We assess our approach on a problem of infrastructure and production planning in offshore
oil fields.