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29
Distributionally robust optimization and its tractable approximations
 Operations Research
"... In this paper, we focus on a linear optimization problem with uncertainties, having expectations in the objective and in the set of constraints. We present a modular framework to obtain an approximate solution to the problem that is distributionally robust, and more flexible than the standard techni ..."
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Cited by 30 (4 self)
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In this paper, we focus on a linear optimization problem with uncertainties, having expectations in the objective and in the set of constraints. We present a modular framework to obtain an approximate solution to the problem that is distributionally robust, and more flexible than the standard technique of using linear rules. Our framework begins by firstly affinelyextending the set of primitive uncertainties to generate new linear decision rules of larger dimensions, and are therefore more flexible. Next, we develop new piecewiselinear decision rules which allow a more flexible reformulation of the original problem. The reformulated problem will generally contain terms with expectations on the positive parts of the recourse variables. Finally, we convert the uncertain linear program into a deterministic convex program by constructing distributionally robust bounds on these expectations. These bounds are constructed by first using different pieces of information on the distribution of the underlying uncertainties to develop separate bounds, and next integrating them into a combined bound that is better than each of the individual bounds.
Design of affine controllers via convex optimization
, 2008
"... Abstract—We consider a discretetime timevarying linear dynamical system, perturbed by process noise, with linear noise corrupted measurements, over a finite horizon. We address the problem of designing a general affine causal controller, in which the control input is an affine function of all prev ..."
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Cited by 14 (1 self)
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Abstract—We consider a discretetime timevarying linear dynamical system, perturbed by process noise, with linear noise corrupted measurements, over a finite horizon. We address the problem of designing a general affine causal controller, in which the control input is an affine function of all previous measurements, in order to minimize a convex objective, in either a stochastic or worstcase setting. This controller design problem is not convex in its natural form, but can be transformed to an equivalent convex optimization problem by a nonlinear change of variables, which allows us to efficiently solve the problem. Our method is related to the classicaldesign procedure for timeinvariant, infinitehorizon linear controller design, and the more recent purified output control method. We illustrate the method with applications to supply chain optimization and dynamic portfolio optimization, and show the method can be combined with model predictive control techniques when perfect state information is available. Index Terms—Affine controller, dynamical system, dynamic linear programming (DLP), linear exponential quadratic Gaussian (LEQG), linear quadratic Gaussian (LQG), model predictive control (MPC), proportionalintegralderivative (PID). I.
P.A.: A hierarchy of nearoptimal policies for multistage adaptive optimization
 IEEE Trans. Autom. Control
"... Abstract In this paper, we propose a new tractable framework for dealing with multistage decision problems affected by uncertainty, applicable to robust optimization and stochastic programming. We introduce a hierarchy of polynomial disturbancefeedback control policies, and show how these can be ..."
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Cited by 13 (3 self)
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Abstract In this paper, we propose a new tractable framework for dealing with multistage decision problems affected by uncertainty, applicable to robust optimization and stochastic programming. We introduce a hierarchy of polynomial disturbancefeedback control policies, and show how these can be computed by solving a single semidefinite programming problem. The approach yields a hierarchy parameterized by a single variable (the degree of the polynomial policies), which controls the tradeoff between the quality of the objective function value and the computational requirements. We evaluate our framework in the context of two classical inventory management applications, in which very strong numerical performance is exhibited, at relatively modest computational expense.
Uncertain Linear Programs: Extended Affinely Adjustable Robust Counterparts
, 2009
"... In this paper, we introduce the extended affinely adjustable robust counterpart to modeling and solving multistage uncertain linear programs with fixed recourse. Our approach first reparameterizes the primitive uncertainties and then applies the affinely adjustable robust counterpart proposed in the ..."
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Cited by 8 (1 self)
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In this paper, we introduce the extended affinely adjustable robust counterpart to modeling and solving multistage uncertain linear programs with fixed recourse. Our approach first reparameterizes the primitive uncertainties and then applies the affinely adjustable robust counterpart proposed in the literature, in which recourse decisions are restricted to be linear in terms of the primitive uncertainties. We propose a special case of the extended affinely adjustable robust counterpart—the splittingbased extended affinely adjustable robust counterpart—and illustrate both theoretically and computationally that the potential of the affinely adjustable robust counterpart method is well beyond the one presented in the literature. Similar to the affinely adjustable robust counterpart, our approach ends up with deterministic optimization formulations that are tractable and scalable to multistage problems.
Robust Local Search for Solving RCPSP/max with Durational Uncertainty
"... Scheduling problems in manufacturing, logistics and project management have frequently been modeled using the framework of Resource Constrained Project Scheduling Problems with minimum and maximum time lags (RCPSP/max). Due to the importance of these problems, providing scalable solution schedules f ..."
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Cited by 6 (2 self)
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Scheduling problems in manufacturing, logistics and project management have frequently been modeled using the framework of Resource Constrained Project Scheduling Problems with minimum and maximum time lags (RCPSP/max). Due to the importance of these problems, providing scalable solution schedules for RCPSP/max problems is a topic of extensive research. However, all existing methods for solving RCPSP/max assume that durations of activities are known with certainty, an assumption that does not hold in real world scheduling problems where unexpected external events such as manpower availability, weather changes, etc. lead to delays or advances in completion of activities. Thus, in this paper, our focus is on providing a scalable method for solving RCPSP/max problems with durational uncertainty. To that end, we introduce the robust local search
Dynamic Portfolio Choice with Linear Rebalancing Rules
, 2012
"... We consider a broad class of dynamic portfolio optimization problems that allow for complex models of return predictability, transaction costs, trading constraints, and risk considerations. Determining an optimal policy in this general setting is almost always intractable. We propose a class of line ..."
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Cited by 6 (1 self)
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We consider a broad class of dynamic portfolio optimization problems that allow for complex models of return predictability, transaction costs, trading constraints, and risk considerations. Determining an optimal policy in this general setting is almost always intractable. We propose a class of linear rebalancing rules, and describe an efficient computational procedure to optimize with this class. We illustrate this method in the context of portfolio execution, and show that it achieves near optimal performance.
Generalized Decision Rule Approximations for Stochastic Programming via Liftings
, 2013
"... Stochastic programming provides a versatile framework for decisionmaking under uncertainty, but the resulting optimization problems can be computationally demanding. It has recently been shown that primal and dual linear decision rule approximations can yield tractable upper and lower bounds on the ..."
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Cited by 5 (3 self)
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Stochastic programming provides a versatile framework for decisionmaking under uncertainty, but the resulting optimization problems can be computationally demanding. It has recently been shown that primal and dual linear decision rule approximations can yield tractable upper and lower bounds on the optimal value of a stochastic program. Unfortunately, linear decision rules often provide crude approximations that result in loose bounds. To address this problem, we propose a lifting technique that maps a given stochastic program to an equivalent problem on a higherdimensional probability space. We prove that solving the lifted problem in primal and dual linear decision rules provides tighter bounds than those obtained from applying linear decision rules to the original problem. We also show that there is a onetoone correspondence between linear decision rules in the lifted problem and families of nonlinear decision rules in the original problem. Finally, we identify structured liftings that give rise to highly flexible piecewise linear and nonlinear decision rules, and we assess their performance in the context of a dynamic production planning problem. 1
Robust Optimization Made Easy with ROME
, 2009
"... We introduce ROME, an algebraic modeling toolbox for modeling a class of robust optimization problems. ROME serves as an intermediate layer between the modeler and optimization solver engines, allowing modelers to express robust optimization problems in a mathematically meaningful way. In this paper ..."
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Cited by 3 (0 self)
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We introduce ROME, an algebraic modeling toolbox for modeling a class of robust optimization problems. ROME serves as an intermediate layer between the modeler and optimization solver engines, allowing modelers to express robust optimization problems in a mathematically meaningful way. In this paper, we discuss how ROME can be used to model (1) a serviceconstrained robust inventory management problem, (2) a project crashing problem, and (3) a robust portfolio optimization problem. Through these modeling examples, we highlight the key features of ROME which allows it to expedite the modeling and subsequent numerical analysis of robust optimization problems. ROME is freely distributed for academic use from www.robustopt.com.
Dynamic Stochastic Orienteering Problems for RiskAware Applications
 In UAI ’2012: Proceedings of Conf. on Uncertainty in AI
, 2012
"... Orienteering problems (OPs) are a variant of the wellknown prizecollecting traveling salesman problem, where the salesman needs to choose a subset of cities to visit within a given deadline. OPs and their extensions with stochastic travel times (SOPs) have been used to model vehicle routing proble ..."
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Cited by 3 (3 self)
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Orienteering problems (OPs) are a variant of the wellknown prizecollecting traveling salesman problem, where the salesman needs to choose a subset of cities to visit within a given deadline. OPs and their extensions with stochastic travel times (SOPs) have been used to model vehicle routing problems and tourist trip design problems. However, they suffer from two limitations – travel times between cities are assumed to be time independent and the route provided is independent of the risk preference (with respect to violating the deadline) of the user. To address these issues, we make the following contributions: We introduce (1) a dynamic SOP (DSOP) model, which is an extension of SOPs with dynamic (timedependent) travel times; (2) a risksensitive criterion to allow for different risk preferences; and (3) a local search algorithm to solve DSOPs with this risksensitive criterion. We evaluated our algorithms on a realworld dataset for a theme park navigation problem as well as synthetic datasets employed in the literature. 1