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27
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 21 (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
 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.
On the Power of Robust Solutions in TwoStage Stochastic and Adaptive Optimization Problems
"... informs doi 10.1287/moor.1090.0440 ..."
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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Cited by 1 (0 self)
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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.
Robust Optimization with Multiple Ranges: Theory and Application to R & D Project Selection. Submitted to European
 Journal of Operational Research, Available at http://www.optimizationonline.org/ Ierapetritou, M.G.; Wu, D.; Vin, J.; Sweeny P.; Chigirinskiy M
, 2010
"... We present a robust optimization approach when the uncertainty in objective coefficients is described using multiple ranges for each coefficient. This setting arises when the value of the uncertain coefficients, such as cash flows, depends on an underlying random variable, such as the effectiveness ..."
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We present a robust optimization approach when the uncertainty in objective coefficients is described using multiple ranges for each coefficient. This setting arises when the value of the uncertain coefficients, such as cash flows, depends on an underlying random variable, such as the effectiveness of a new drug. Traditional robust optimization with a single range per coefficient would require very large ranges in this case and lead to overly conservative results. In our approach, the decisionmaker limits the number of coefficients that fall within each range; he can also limit the number of coefficients that deviate from their nominal value in a given range. Modeling multiple ranges requires the use of binary variables in the uncertainty set. We show how to address this issue to develop tractable reformulations and apply our approach to a R&D project selection problem when cash flows are uncertain. Furthermore, we develop a robust ranking heuristic, where the project manager ranks the projects according to densities (ratio of cash flows to development costs) or Net Present Values, while incorporating the budgets of uncertainty but without requiring any optimization procedure. While both densitybased and NPVbased ranking heuristics perform very well in experiments, the NPVbased heuristic performs better; in particular, it finds the truly optimal solution more often.
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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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.
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
2Stage Robust MILP with continuous recourse variables
"... 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 un ..."
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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
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 un ..."
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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