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Publication# Distributionally Robust Optimization with Markovian Data

Résumé

We study a stochastic program where the probability distribution of the uncertain problem parameters is unknown and only indirectly observed via finitely many correlated samples generated by an unknown Markov chain with d states. We propose a data-driven distributionally robust optimization model to estimate the problem’s objective function and optimal solution. By leveraging results from large deviations theory, we derive statistical guarantees on the quality of these estimators. The underlying worst-case expectation problem is nonconvex and involves O(d^2) decision variables. Thus, it cannot be solved efficiently for large d. By exploiting the structure of this prob- lem, we devise a customized Frank-Wolfe algorithm with convex direction-finding subproblems of size O(d). We prove that this algorithm finds a stationary point efficiently under mild conditions. The efficiency of the method is predicated on a dimensionality reduction enabled by a dual reformulation. Numerical experiments indicate that our approach has better computational and statistical properties than the state-of-the-art methods.

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Publications associées (31)

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Concepts associés (19)

Stochastic programming

In the field of mathematical optimization, stochastic programming is a framework for modeling optimization problems that involve uncertainty. A stochastic program is an optimization problem in which

Optimisation robuste

L'optimisation robuste est une branche de l'optimisation mathématique qui cherche à résoudre un problème d'optimisation en prenant en compte les différentes sources d'incertitude de celui-ci.
Hi

Point stationnaire

350px|thumb|right|Les points stationnaires de la fonction sont marquées par des ronds rouges. Dans ce cas, ce sont des extrema locaux. Les carrés bleus désignent les points d'inflexion.
En analyse rée

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.

We have developed a new derivative-free algorithm based on Radial Basis Functions (RBFs). Derivative-free optimization is an active field of research and several algorithms have been proposed recently. Problems of this nature in the industrial setting are quite frequent. The reason is that in a number of applications the optimization process contains simulation packages which are treated as black boxes. The development of our own algorithm was originally motivated by an application in biomedical imaging: the medical image registration problem. The particular characteristics of this problem have incited us to develop a new optimization algorithm based on trust-region methods. However it has been designed to be generic and to be applied to a wide range of problems. The main originality of our approach is the use of RBFs to build the models. In particular we have adapted the existing theory based on quadratic models to our own models and developed new procedures especially designed for models based on RBFs. We have tested our algorithm called BOOSTERS against state-of-the-art methods (UOBYQA, NEWUOA, DFO). On the medical image registration problem, BOOSTERS appears to be the method of choice. The tests on problems from the CUTEr collection show that BOOSTERS is comparable to, but not better than other methods on small problems (size 2-20). It is performing very well for medium size problems (20-80). Moreover, it is able to solve problems of dimension 200, which is considered very large in derivative-free optimization. We have also developed a new class of algorithms combining the robustness of derivative-free algorithms with the faster rate of convergence characterizing Newtonlike-methods. In fact, they define a new class of algorithms lying between derivative-free optimization and quasi-Newton methods. These algorithms are built on the skeleton of our derivative-free algorithm but they can incorporate the gradient when it is available. They can be interpreted as a way of doping derivative-free algorithms with derivatives. If the derivatives are available at each iteration, then our method can be seen as an alternative to quasi-Newton methods. At the opposite, if the derivatives are never evaluated, then the algorithm is totally similar to BOOSTERS. It is a very interesting alternative to existing methods for problems whose objective function is expensive to evaluate and when the derivatives are not available. In this situation, the gradient can be approximated by finite differences and its costs corresponds to n additional function evaluations assuming that Rn is the domain of definition of the objective function. We have compared our method with CFSQP and BTRA, two gradient-based algorithms, and the results show that our doped method performs best. We have also a theoretical analysis of the medical image registration problem based on maximization of mutual information. Most of the current research in this field is concentrated on registration based on nonlinear image transformation. However, little attention has been paid to the theoretical properties of the optimization problem. In our analysis, we focus on the continuity and the differentiability of the objective function. We show in particular that performing a registration without extension of the reference image may lead to discontinuities in the objective function. But we demonstrate that, under some mild assumptions, the function is differentiable almost everywhere. Our analysis is important from an optimization point of view and conditions the choice of a solver. The usual practice is to use generic optimization packages without worrying about the differentiability of the objective function. But the use of gradient-based methods when the objective function is not differentiable may result in poor performance or even in absence of convergence. One of our objectives with this analysis is also that practitioners become aware of these problems and to propose them new algorithms having a potential interest for their applications.

Stochastic programming and robust optimization are disciplines concerned with optimal decision-making under uncertainty over time. Traditional models and solution algorithms have been tailored to problems where the order in which the uncertainties unfold is independent of the controller actions. Nevertheless, in numerous real-world decision problems, the time of information discovery can be influenced by the decision maker, and uncertainties only become observable following an (often costly) investment. Such problems can be formulated as mixed-binary multi-stage stochastic programs with decision-dependent non-anticipativity constraints. Unfortunately, these problems are severely computationally intractable. We propose an approximation scheme for multi-stage problems with decision-dependent information discovery which is based on techniques commonly used in modern robust optimization. In particular, we obtain a conservative approximation in the form of a mixed-binary linear program by restricting the spaces of measurable binary and real-valued decision rules to those that are representable as piecewise constant and linear functions of the uncertain parameters, respectively. We assess our approach on a problem of infrastructure and production planning in offshore oil fields from the literature.