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Publication# A General Framework for Optimal Data-Driven Optimization

Résumé

We propose a statistically optimal approach to construct data-driven decisions for stochastic optimization problems. Fundamentally, a data-driven decision is simply a function that maps the available training data to a feasible action. It can always be expressed as the minimizer of a surrogate optimization model constructed from the data. The quality of a data-driven decision is measured by its out-of-sample risk. An additional quality measure is its out-of-sample disappointment, which we define as the probability that the out-of-sample risk exceeds the optimal value of the surrogate optimization model. The crux of data-driven optimization is that the data-generating probability measure is unknown. An ideal data-driven decision should therefore minimize the out-of-sample risk simultaneously with respect to every conceivable probability measure (and thus in particular with respect to the unknown true measure). Unfortunately, such ideal data-driven decisions are generally unavailable. This prompts us to seek data-driven decisions that minimize the out-of-sample risk subject to an upper bound on the out-of-sample disappointment---again simultaneously with respect to every conceivable probability measure. We prove that such Pareto-dominant data-driven decisions exist under conditions that allow for interesting applications: the unknown data-generating probability measure must belong to a parametric ambiguity set, and the corresponding parameters must admit a sufficient statistic that satisfies a large deviation principle. If these conditions hold, we can further prove that the surrogate optimization model generating the optimal data-driven decision must be a distributionally robust optimization problem constructed from the sufficient statistic and the rate function of its large deviation principle. This shows that the optimal method for mapping data to decisions is, in a rigorous statistical sense, to solve a distributionally robust optimization model. Maybe surprisingly, this result holds irrespective of whether the original stochastic optimization problem is convex or not and holds even when the training data is non-i.i.d. As a byproduct, our analysis reveals how the structural properties of the data-generating stochastic process impact the shape of the ambiguity set underlying the optimal distributionally robust optimization model.

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Stochastic optimization is a popular modeling paradigm for decision-making under uncertainty and has a wide spectrum of applications in management science, economics and engineering. However, the stochastic optimization models one faces in practice are intractable, and numerical solutions necessitate approximations. The mainstream approach for making a stochastic optimization model amenable to numerical solution is to discretize the probability distribution of the uncertain problem parameters. However, both the accuracy of the approximation as well as the computational burden of solving the approximate problem scale with the number of scenarios of the approximate distribution. An effective means to ease the computational burden is to use scenario reduction, which replaces an accurate initial distribution accommodating many scenarios with a simpler distribution supported on only few scenarios that is close to the initial distribution with respect to a probability metric.
Using the Wasserstein distance as measure of proximity between distributions, we provide new insights into the fundamental limitations of scenario reduction, and we propose the first polynomial-time constant-factor approximations for a popular scenario reduction problem from the literature. As scenario reduction is equivalent to clustering, it suffers from two well-known shortcomings. Namely, it suffers from outlier sensitivity and may produce highly unbalanced clusters. To mitigate both shortcomings, we formulate a joint outlier detection and clustering problem, where the clusters must satisfy certain cardinality constraints. We cast this problem as a mixed-integer linear program (MILP) that admits tractable semidefinite and linear programming relaxations. We propose deterministic rounding schemes that transform the relaxed solutions to feasible solutions for the MILP. We also prove that these solutions are optimal in the MILP if a cluster separation condition holds. Finally, we develop a highly efficient scenario reduction method for a large-scale hydro scheduling problem. Specifically, we study the optimal operation of a fleet of interconnected hydropower plants that sell energy on both the spot and the reserve markets, and we propose a two-layer stochastic programming framework for its solution. The outer layer problem (the planner's problem) optimizes the end-of-day reservoir filling levels over one year, whereas the inner layer problem (the trader's problem) selects optimal hourly market bids within each day. We prove that the trader's problem admits a scenario reduction that dramatically reduces its complexity without loss of optimality, which in turn facilitates an efficient approximation of the planner's problem.

Dynamic optimization problems affected by uncertainty are ubiquitous in many application domains. Decision makers typically model the uncertainty through random variables governed by a probability distribution. If the distribution is precisely known, then the emerging optimization problems constitute stochastic programs or chance constrained programs. On the other hand, if the distribution is at least partially unknown, then the emanating optimization problems represent robust or distributionally robust optimization problems. In this thesis, we leverage techniques from stochastic and distributionally robust optimization to address complex problems in finance, energy systems management and, more abstractly, applied probability. In particular, we seek to solve uncertain optimization problems where the prior distributional information includes only the first and the second moments (and, sometimes, the support). The main objective of the thesis is to solve large instances of practical optimization problems. For this purpose, we develop complexity reduction and decomposition schemes, which exploit structural symmetries or multiscale properties of the problems at hand in order to break them down into smaller and more tractable components. In the first part of the thesis we study the growth-optimal portfolio, which maximizes the expected log-utility over a single investment period. In a classical stochastic setting, this portfolio is known to outperform any other portfolio with probability 1 in the long run. In the short run, however, it is notoriously volatile. Moreover, its performance suffers in the presence of distributional ambiguity. We design fixed-mix strategies that offer similar performance guarantees as the classical growth-optimal portfolio but for a finite investment horizon. Moreover, the proposed performance guarantee remains valid for any asset return distribution with the same mean and covariance matrix. These results rely on a Taylor approximation of the terminal logarithmic wealth that becomes more accurate as the rebalancing frequency is increased. In the second part of the thesis, we demonstrate that such a Taylor approximation is in fact not necessary. Specifically, we derive sharp probability bounds on the tails of a product of non-negative random variables. These generalized Chebyshev bounds can be computed numerically using semidefinite programming--in some cases even analytically. Similar techniques can also be used to derive multivariate Chebyshev bounds for sums, maxima, and minima of random variables. In the final part of the thesis, we consider a multi-market reservoir management problem. The eroding peak/off-peak spreads on European electricity spot markets imply reduced profitability for the hydropower producers and force them to participate in the balancing markets. This motivates us to propose a two-layer stochastic programming model for the optimal operation of a cascade of hydropower plants selling energy on both spot and balancing markets. The planning problem optimizes the reservoir management over a yearly horizon with weekly granularity, and the trading subproblems optimize the market transactions over a weekly horizon with hourly granularity. We solve both the planning and trading problems in linear decision rules, and we exploit the inherent parallelizability of the trading subproblems to achieve computational tractability.

Dynamic stochastic optimization problems with a large (possibly infinite) number of decision stages and high-dimensional state vectors are inherently difficult to solve. In fact, scenario tree-based algorithms are unsuitable for problems with many stages, while dynamic programming-type techniques are unsuitable for problems with many state variables. This paper proposes a stage aggregation scheme for stochastic optimization problems in continuous time, thus having an extremely large (i.e., uncountable) number of decision stages. By perturbing the underlying data and information processes, we construct two approximate problems that provide bounds on the optimal value of the original problem. Moreover, we prove that the gap between the bounds converges to zero as the stage aggregation is refined. If massive aggregation of stages is possible without sacrificing too much accuracy, the aggregate approximate problems can be addressed by means of scenario tree-based methods. The suggested approach applies to problems that exhibit randomness in the objective and the constraints, while the constraint functions are required to be additively separable in the decision variables and random parameters.

2009