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34
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 60 (4 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.
A simplex based algorithm to solve separated continuous linear programs
 Mathematical Programming
, 2008
"... We consider the separated continuous linear programming problem with linear data. We characterize the form of its optimal solution, and present an algorithm which solves it in a finite number of steps, using simplex pivot iterations. 1 ..."
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Cited by 30 (5 self)
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We consider the separated continuous linear programming problem with linear data. We characterize the form of its optimal solution, and present an algorithm which solves it in a finite number of steps, using simplex pivot iterations. 1
A Semidefinite Programming Approach to Optimal Moment Bounds for Convex Classes of Distributions
, 2005
"... We provide an optimization framework for computing optimal upper and lower bounds on functional expectations of distributions with special properties, given moment constraints. Bertsimas and Popescu have already shown how to obtain optimal moment inequalities for arbitrary distributions via semidefi ..."
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Cited by 24 (0 self)
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We provide an optimization framework for computing optimal upper and lower bounds on functional expectations of distributions with special properties, given moment constraints. Bertsimas and Popescu have already shown how to obtain optimal moment inequalities for arbitrary distributions via semidefinite programming. These bounds are not sharp if the underlying distributions possess additional structural properties, including symmetry, unimodality, convexity or smoothness. For convex distribution classes that are in some sense generated by an appropriate parametric family, we use conic duality to show how optimal moment bounds can be efficiently computed as semidefinite programs. In particular, we obtain generalizations of Chebyshev’s inequality for symmetric and unimodal distributions, and provide numerical calculations to compare these bounds given higher order moments. We also extend these results for multivariate distributions.
FOUR PROOFS OF GITTINS’ MULTIARMED BANDIT THEOREM
 APPLIED PROBABILITY TRUST
, 1999
"... We survey four proofs that the Gittins index priority rule is optimal for alternative bandit processes. These include Gittins’ original exchange argument, Weber’s prevailing charge argument, Whittle’s Lagrangian dual approach, and a proof based on generalized conservation laws and LP duality. ..."
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Cited by 17 (0 self)
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We survey four proofs that the Gittins index priority rule is optimal for alternative bandit processes. These include Gittins’ original exchange argument, Weber’s prevailing charge argument, Whittle’s Lagrangian dual approach, and a proof based on generalized conservation laws and LP duality.
Space tensor conic programming
, 2009
"... Space tensors appear in physics and mechanics, and they are real physical entities. Mathematically, they are tensors in the threedimensional Euclidean space. In the research of diffusion magnetic resonance imaging, convex optimization problems are formed where higher order positive semidefinite sp ..."
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Cited by 7 (5 self)
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Space tensors appear in physics and mechanics, and they are real physical entities. Mathematically, they are tensors in the threedimensional Euclidean space. In the research of diffusion magnetic resonance imaging, convex optimization problems are formed where higher order positive semidefinite space tensors are involved. In this short paper, we investigate these problems from the viewpoint of conic linear programming (CLP). We characterize the dual cone of the positive semidefinite space tensor cone, and study the CLP formulation and the duality of the positive semidefinite space tensor conic programming problem.
Bounding Option Prices of MultiAssets: A Semidefinite Programming Approach
"... Recently, semidefinite programming has been used to bound the price of a singleasset European call option at a fixed time. Given the first n moments, a tight bound can be obtained by solving a single semidefinite programming problem of dimension n + 1. In this paper, we study the multiasset case, ..."
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Cited by 4 (0 self)
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Recently, semidefinite programming has been used to bound the price of a singleasset European call option at a fixed time. Given the first n moments, a tight bound can be obtained by solving a single semidefinite programming problem of dimension n + 1. In this paper, we study the multiasset case, which is generally more practical than the singleasset case. We construct a sequence of semidefinite programming relaxations. As the dimension of the semidefinite relaxations increases, the bound becomes more accurate and converges to the tight bound. Some numerical results are reported to illustrate the method. Key words: Bound of option price, semidefinite programming relaxation, the moment problem.
Robust and stochastically weighted multiobjective optimization models and reformulations
, 2011
"... We introduce and study a family of models for multiexpert multiobjective/criteria decision making. These models use a concept of weight robustness to generate a risk averse decision. In particular, the multiexpert multicriteria robust weighted sum approach (McRow) introduced in this paper identi ..."
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Cited by 4 (0 self)
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We introduce and study a family of models for multiexpert multiobjective/criteria decision making. These models use a concept of weight robustness to generate a risk averse decision. In particular, the multiexpert multicriteria robust weighted sum approach (McRow) introduced in this paper identifies a (robust) Pareto optimum decision that minimizes the worst case weighted sum of objectives over a given weight region. The corresponding objective value, called the robustvalue of a decision, is shown to be increasing and concave in the weight set. Compact reformulations of the models with polyhedral and conic descriptions of the weight regions. The McRow model is developed further for stochastic multiexpert multicriteria decision making by allowing ambiguity or randomness in the weight region as well as the objective functions. The properties of the proposed approach is illustrated with a few examples. The usefulness of the stochastic (McRow) model is demonstrated using a disaster planning example and an agriculture revenue management example.
Separated Continuous Conic Programming: Strong Duality and an Approximation Algorithm ∗
, 2006
"... Motivated by recent applications in robust optimization and in signconstrained linearquadratic control, we study in this paper a new class of optimization problems called separated continuous conic programming (SCCP). Focusing on a symmetric primaldual pair, we develop a strong duality theory for ..."
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Motivated by recent applications in robust optimization and in signconstrained linearquadratic control, we study in this paper a new class of optimization problems called separated continuous conic programming (SCCP). Focusing on a symmetric primaldual pair, we develop a strong duality theory for the SCCP. Our idea is to use discretization to connect the SCCP and its dual to two ordinary conic programs. We show if the latter are strongly feasible and with finite optimal values, a condition that is readily verifiable, then the strong duality holds for the SCCP. This approach also leads to a polynomialtime approximation algorithm that solves the SCCP to any required accuracy.
Preprocessing and Regularization for Degenerate Semidefinite Programs
, 2013
"... This paper presentsa backward stable preprocessing technique for (nearly) illposed semidefinite programming, SDP, problems, i.e., programs for which the Slater constraint qualification, existence of strictly feasible points, (nearly) fails. Current popular algorithms for semidefinite programming r ..."
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Cited by 3 (0 self)
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This paper presentsa backward stable preprocessing technique for (nearly) illposed semidefinite programming, SDP, problems, i.e., programs for which the Slater constraint qualification, existence of strictly feasible points, (nearly) fails. Current popular algorithms for semidefinite programming rely on primaldual interiorpoint, pd ip methods. These algorithms require the Slater constraint qualification for both the primal and dual problems. This assumption guarantees the existence of Lagrange multipliers, wellposedness of the problem, and stability of algorithms. However, there are many instances of SDPs where the Slater constraint qualification fails or nearly fails. Our backward stable preprocessing technique is based on applying the BorweinWolkowicz facial reduction process to find a finite number, k, of rankrevealing orthogonal rotations of the problem. After an appropriate truncation, this results in a smaller, wellposed, nearby problem that satisfies the Robinson constraint qualification, and one that can be solved by standard SDP solvers. The