Decision Rule Approximations for Dynamic Optimization under Uncertainty

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

Confidence Levels for CVaR Risk Measures and Minimax Limits*

Conditional value at risk (CVaR) has been widely used as a risk measure in finance. When the confidence level of CVaR is set close to 1, the CVaR risk measure approximates the extreme (worst scenario) risk measure. In this paper, we present a quantitative analysis of the relationship between the two risk measures and it’s impact on optimal decision making when we wish to minimize the respective risk measures. We also investigate the difference between the optimal solutions to the two optimization problems with identical objective function but under constraints on the two risk measures. We discuss the benefits of a sample average approximation scheme for the CVaR constraints and investigate the convergence of the optimal solution obtained from this scheme as the sample size increases. We use some portfolio optimization problems to investigate teh performance of the CVaR approximation approach. Our numerical results demonstrate how reducing the confidence level can lead to a better overall performance

A Planner-Trader Decomposition for Multi-Market Hydro Scheduling

Peak/off-peak spreads on European electricity forward and spot markets are eroding due to the ongoing nuclear phaseout and the steady growth in photovoltaic capacity. The reduced profitability of peak/off-peak arbitrage forces hydropower producers to recover part of their original profitability on the reserve markets. We propose a bi-layer stochastic programming framework for the optimal operation of a fleet of interconnected hydropower plants that sells energy on both the spot and the reserve markets. The outer layer (the planner's problem) optimizes end-of-day reservoir filling levels over one year, whereas the inner layer (the trader's problem) selects optimal hourly market bids within each day. Using an information restriction whereby the planner prescribes the end-of-day reservoir targets one day in advance, we prove that the trader's problem simplifies from an infinite-dimensional stochastic program with 25 stages to a finite two-stage stochastic program with only two scenarios. Substituting this reformulation back into the outer layer and approximating the reservoir targets by affine decision rules allows us to simplify the planner's problem from an infinite-dimensional stochastic program with 365 stages to a two-stage stochastic program that can conveniently be solved via the sample average approximation. Numerical experiments based on a cascade in the Salzburg region of Austria demonstrate the effectiveness of the suggested framework

The Decision Rule Approach to Optimisation under Uncertainty: Theory and Applications

Optimisation under uncertainty has a long and distinguished history in operations research. Decision-makers realised early on that the failure to account for uncertainty in optimisation problems can lead to substantial unexpected losses or even infeasible solutions. Therefore, approximating the uncertain parameters by their average or nominal values may result in decisions that perform poorly in scenarios that deviate from the average. For the last sixty years, scenario tree-based stochastic programming has been the method of choice for solving optimisation problems affected by parameter uncertainty. This method approximates the random problem parameters by finite scenarios that can be arranged as a tree. Unfortunately, this approximation suffers from a curse of dimensionality: the tree needs to branch whenever new uncertainties are revealed, and thus its size grows exponentially with the number of decision stages. It has recently been argued that stochastic programs can quite generally be made tractable by restricting the space of recourse decisions to those that exhibit a linear data dependence. An attractive feature of this linear decision rule approximation is that it typically leads to polynomial-time solution schemes. Unfortunately, the simple structure of linear decision rules sacrifices optimality in return for scalability. The worst-case performance of linear decision rules is in fact rather disappointing. When applied to two-stage robust optimisation problems with m linear constraints, the underlying worst-case approximation ratio has been shown to be of the order O(√m). Therefore, in this thesis we endeavour to construct efficiently computable instance-wise bounds on the loss of optimality incurred by the linear decision rule approximation. The contributions of this thesis are as follows. (i)We develop an efficient algorithm for assessing the loss of optimality incurred by the linear decision rule approximation. The key idea is to apply the linear decision rule restriction not only to the primal but also to a dual version of the stochastic program. Since both problems share a similar structure, both problems can be solved in polynomial-time. The gap between their optimal values estimates the loss of optimality incurred by the linear decision rule approximation. (ii) We design an improved approximation based on non-linear decision rules, which can be useful if the optimality gap of the linear decision rules is deemed unacceptably high. The idea takes advantage of the fact that one can always map a linearly parameterised non-linear function into a higher dimensional space, where it can be represented as a linear function. This allows us to utilise the machinery developed for linear decision rules to produce superior quality approximations that can be obtained in polynomial time. (iii) We assess the performance of the approximations developed in two operations management problems: a production planning problem and a supply chain design problem. We show that near-optimal solutions can be found in problem instances with many stages and random parameters. We additionally compare the quality of the decision rule approximation with classical approximation techniques. (iv) We develop a systematic approach to reformulate multi-stage stochastic programs with a large (possibly infinite) number of stages as static robust optimisation problem that can be solved with a constraint sampling technique. The method is motivated via an investment planning problem in the electricity industry