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20
Distributionally Robust Optimization under Moment Uncertainty with Application to DataDriven Problems
"... Stochastic programs can effectively describe the decisionmaking problem in an uncertain environment. Unfortunately, such programs are often computationally demanding to solve. In addition, their solutions can be misleading when there is ambiguity in the choice of a distribution for the random param ..."
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Cited by 20 (2 self)
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Stochastic programs can effectively describe the decisionmaking problem in an uncertain environment. Unfortunately, such programs are often computationally demanding to solve. In addition, their solutions can be misleading when there is ambiguity in the choice of a distribution for the random parameters. In this paper, we propose a model describing one’s uncertainty in both the distribution’s form (discrete, Gaussian, exponential, etc.) and moments (mean and covariance). We demonstrate that for a wide range of cost functions the associated distributionally robust stochastic program can be solved efficiently. Furthermore, by deriving new confidence regions for the mean and covariance of a random vector, we provide probabilistic arguments for using our model in problems that rely heavily on historical data. This is confirmed in a practical example of portfolio selection, where our framework leads to better performing policies on the “true” distribution underlying the daily return of assets.
Cuttingset methods for robust convex optimization with pessimizing oracles,” Optim
 Department of Electrical and Computer Engineering, University of California, San Diego. From
, 2011
"... We consider a general worstcase robust convex optimization problem, with arbitrary dependence on the uncertain parameters, which are assumed to lie in some given set of possible values. We describe a general method for solving such a problem, which alternates between optimization and worstcase ana ..."
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Cited by 10 (5 self)
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We consider a general worstcase robust convex optimization problem, with arbitrary dependence on the uncertain parameters, which are assumed to lie in some given set of possible values. We describe a general method for solving such a problem, which alternates between optimization and worstcase analysis. With exact worstcase analysis, the method is shown to converge to a robust optimal point. With approximate worstcase analysis, which is the best we can do in many practical cases, the method seems to work very well in practice, subject to the errors in our worstcase analysis. We give variations on the basic method that can give enhanced convergence, reduce data storage, or improve other algorithm properties. Numerical simulations suggest that the method finds a quite robust solution within a few tens of steps; using warmstart techniques in the optimization steps reduces the overall effort to a modest multiple of solving a nominal problem, ignoring the parameter variation. The method is illustrated with several application examples. Key words. Robust optimization, cuttingset methods, semiinfinite programming, minimax optimization, games.
On the Power of Robust Solutions in TwoStage Stochastic and Adaptive Optimization Problems
"... informs doi 10.1287/moor.1090.0440 ..."
Layered Formulation for the Robust Vehicle Routing Problem with Time Windows
, 2013
"... Abstract. This paper studies the vehicle routing problem with time windows where travel times are uncertain and belong to a predetermined polytope. The objective of the problem is to find a set of routes that services all nodes of the graph and that are feasible for all values of the travel times in ..."
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Cited by 1 (0 self)
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Abstract. This paper studies the vehicle routing problem with time windows where travel times are uncertain and belong to a predetermined polytope. The objective of the problem is to find a set of routes that services all nodes of the graph and that are feasible for all values of the travel times in the uncertainty polytope. The problem is motivated by maritime transportation where delays are frequent and must be taken into account. We present an extended formulation for the vehicle routing problem with time windows that allows us to apply the classical (static) robust programming approach to the problem. The formulation is based on a layered representation of the graph, which enables to track the position of each arc in its route. We test our formulation on a test bed composed of maritime transportation instances.
A Geometric Characterization of the Power of Finite Adaptability in Multistage Stochastic and Adaptive Optimization
"... In this paper, we show a significant role that geometric properties of uncertainty sets, such as symmetry, play in determining the power of robust and finitely adaptable solutions in multistage stochastic and adaptive optimization problems. We consider a fairly general class of multistage mixed in ..."
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In this paper, we show a significant role that geometric properties of uncertainty sets, such as symmetry, play in determining the power of robust and finitely adaptable solutions in multistage stochastic and adaptive optimization problems. We consider a fairly general class of multistage mixed integer stochastic and adaptive optimization problems and propose a good approximate solution policy with performance guarantees that depend on the geometric properties of the uncertainty sets. In particular, we show that a class of finitely adaptable solutions is a good approximation for both the multistage stochastic as well as the adaptive optimization problem. A finitely adaptable solution generalizes the notion of a static robust solution and specifies a small set of solutions for each stage and the solution policy implements the best solution from the given set depending on the realization of the uncertain parameters in past stages. Therefore, it is a tractable approximation to a fullyadaptable solution for the multistage problems. To the best of our knowledge, these are the first approximation results for the multistage problem in such generality. Moreover, the results and the proof techniques are quite general and also extend to include important constraints such as integrality and linear conic constraints.
Approximation Algorithms for Offline Riskaverse Combinatorial Optimization
, 2010
"... We consider generic optimization problems that can be formulated as minimizing the cost of a feasible solution w T x over a combinatorial feasible set F ⊂ {0, 1} n. For these problems we describe a framework of riskaverse stochastic problems where the cost vector W has independent random components ..."
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We consider generic optimization problems that can be formulated as minimizing the cost of a feasible solution w T x over a combinatorial feasible set F ⊂ {0, 1} n. For these problems we describe a framework of riskaverse stochastic problems where the cost vector W has independent random components, unknown at the time of solution. A natural and important objective that incorporates risk in this stochastic setting is to look for a feasible solution whose stochastic cost has a small tail or a small convex combination of mean and standard deviation. Our models can be equivalently reformulated as nonconvex programs for which no efficient algorithms are known. In this paper, we make progress on these hard problems. Our results are several efficient generalpurpose approximation schemes. They use as a blackbox (exact or approximate) the solution to the underlying deterministic problem and thus immediately apply to arbitrary combinatorial problems. For example, from an available δapproximation algorithm to the linear problem, we construct a δ(1 + ǫ)approximation algorithm for the stochastic problem, which invokes the linear algorithm only a logarithmic number of times in the problem input (and polynomial in 1 ǫ), for any desired accuracy level ǫ> 0. The algorithms are based on a geometric analysis of the curvature and approximability of the nonlinear level sets of the objective functions. 1
The Influence of Operational Cost on Estimation
"... This work concerns the way that statistical models are used to make decisions. In particular, we aim to merge the way estimation algorithms are designed with how they are used for a subsequent task. Our methodology considers the operational cost of carrying out a policy, based on a predictive model. ..."
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This work concerns the way that statistical models are used to make decisions. In particular, we aim to merge the way estimation algorithms are designed with how they are used for a subsequent task. Our methodology considers the operational cost of carrying out a policy, based on a predictive model. The operational cost becomes a regularization term in the learning algorithm’s objective function, allowing either an optimistic or pessimistic view of possible costs. Limiting the operational cost reduces the hypothesis space for the predictive model, and can thus improve generalization. We show that different types of operational problems can lead to the same type of restriction on the hypothesis space, namely the restriction to an intersection of an ℓq ball with a halfspace. We bound the complexity of such hypothesis spaces by proposing a technique that involves counting integer points in polyhedrons.
RANKING AND SELECTION MEETS ROBUST OPTIMIZATION
"... The objective of ranking and selection is to efficiently allocate an information budget among a set of design alternatives with unknown values in order to maximize the decisionmaker’s chances of discovering the best alternative. The field of robust optimization, however, considers riskaverse decis ..."
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The objective of ranking and selection is to efficiently allocate an information budget among a set of design alternatives with unknown values in order to maximize the decisionmaker’s chances of discovering the best alternative. The field of robust optimization, however, considers riskaverse decision makers who may accept a suboptimal alternative in order to minimize the risk of a worstcase outcome. We bring these two fields together by defining a Bayesian ranking and selection problem with a robust implementation decision. We propose a new simulation allocation procedure that is riskneutral with respect to simulation outcomes, but riskaverse with respect to the implementation decision. We discuss the properties of the procedure and present numerical examples illustrating the difference between the riskaverse problem and the more typical riskneutral problem from the literature. 1
2Stage Robust MILP with continuous
"... We solve a linear robust problem with mixedinteger firststage variables and continuous second stage variables. We consider column wise uncertainty. We first focus on a problem with right handside uncertainty which satisfies a "full recourse property " and a specific definition of the uncertainty. ..."
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We solve a linear robust problem with mixedinteger firststage variables and continuous second stage variables. We consider column wise uncertainty. We first focus on a problem with right handside uncertainty which satisfies a "full recourse property " and a specific definition of the uncertainty. We propose a solution based on a generation constraint algorithm. Then we give several generalizations of the approach: for lefthand side uncertainty, for the cases where the "full recourse property " is not satisfied and for uncertainty sets defined by a polytope. 1
Submitted to Operations Research manuscript (Please, provide the mansucript number!) Robust Queueing Theory
"... We propose an alternative approach for studying queueing systems by employing robust optimization as opposed to stochastic analysis. While traditional stochastic queueing theory relies on Kolmogorov’s axioms of probability and models arrivals and services as renewal processes, we use the limit laws ..."
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We propose an alternative approach for studying queueing systems by employing robust optimization as opposed to stochastic analysis. While traditional stochastic queueing theory relies on Kolmogorov’s axioms of probability and models arrivals and services as renewal processes, we use the limit laws of probability as the axioms of our methodology and model the queueing systems primitives by uncertainty sets. In this framework, we obtain closed form expressions for the steadystate waiting times in multiserver queues with heavytailed arrival and service processes. These expressions are not available under traditional stochastic queueing theory for heavytailed processes, while they lead to the same qualitative insights for independent and identically distributed arrival and service times. We also develop an exact calculus for analyzing a network of queues with multiple servers based on the following key principle: a) the departure from a queue, b) the superposition, and c) the thinning of arrival processes have the same uncertainty set representation as the original arrival processes. We show that our approach, which we call the Robust Queueing Network Analyzer (RQNA) a) yields results with error percentages in single digits (for all experiments we performed) relative to simulation, b) performs significantly better than the Queueing Network Analyzer (QNA) proposed in Whitt (1983), and c) is to a large extent insensitive to the number of servers per queue, the network size, degree of feedback, traffic intensity, and somewhat sensitive to the degree of diversity of external arrival distributions in the network.