Persistence in discrete optimization under data uncertainty

Bertsimas, Dimitris; Natarajan, Karthik; Teo, Chung‐Piaw

doi:10.1007/s10107-006-0710-z

Cited by 58 publications

(52 citation statements)

References 22 publications

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“…One of the earliest results in this context was derived by Kingman [10] for GI/GI/1 queue. For the single server queue, given the mean and variance of the inter-arrival and service times, he derived an upper bound on the expected steady-state waiting time using (3). We outline the proof next since it motivates our approach.…”

Section: Application In Queueing Systemsmentioning

confidence: 97%

A semidefinite optimization approach to the steady-state analysis of queueing systems

Bertsimas

Natarajan

2007

Queueing Syst

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Computing the steady-state distribution in Markov chains for general distributions and general state space is a computationally challenging problem. In this paper, we consider the steady-state stochastic model Wwhere the equality is in distribution. Given partial distributional information on the random variables X, we want to estimate information on the distribution of the steady-state vector W . Such models naturally occur in queueing systems, where the goal is to find bounds on moments of the waiting time under moment information on the service and interarrival times. In this paper, we propose an approach based on semidefinite optimization to find such bounds. We show that the classical Kingman's and Daley's bounds for the expected waiting time in a GI/GI/1 queue are special cases of the proposed approach. We also report computational results in the queueing context that indicate the method is promising.

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Section: Application In Queueing Systemsmentioning

confidence: 97%

A semidefinite optimization approach to the steady-state analysis of queueing systems

Bertsimas

Natarajan

2007

Queueing Syst

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“…(a) The proof of Theorem 1 is inspired from the proofs in Bertsimas et al (2006) and Natarajan et al (2009b) for univariate marginals and Doan and Natarajan (2012) for nonoverlapping multivariate marginals. Theorem 1 extends these results to overlapping multivariate marginals.…”

Section: Then the Fréchet Boundmentioning

confidence: 99%

Robustness to Dependency in Portfolio Optimization Using Overlapping Marginals

Doan

Natarajan

2015

Operations Research

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In this paper, we develop a distributionally robust portfolio optimization model where the robustness is across different dependency structures among the random losses. For a Fréchet class of discrete distributions with overlapping marginals, we show that the distributionally robust portfolio optimization problem is efficiently solvable with linear programming. To guarantee the existence of a joint multivariate distribution consistent with the overlapping marginal information, we make use of a graph theoretic property known as the running intersection property. Building on this property, we develop a tight linear programming formulation to find the optimal portfolio that minimizes the worst-case Conditional Value-at-Risk measure. Lastly, we use a data-driven approach with financial return data to identify the Fréchet class of distributions satisfying the running intersection property and then optimize the portfolio over this class of distributions. Numerical results in two different datasets show that the distributionally robust portfolio optimization model improves on the sample-based approach

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“…Then for instance, one might be interested to determine whether in an optimal solution x * (y) of P y , and for some index i ∈ I, one has x * i (y) = 1 (or x * i (y) = 0) for almost all values of the parameter y ∈ Y. In [3,17] the probability that x * k (y) is 1 is called the persistency of the boolean variable x * k (y) Corollary 3.7. Let K, Y be as in (2.2) and (3.1) respectively.…”

Section: Persistence For Boolean Variablesmentioning

confidence: 99%

A “Joint+Marginal” Approach to Parametric Polynomial Optimization

Lasserre¹

2010

SIAM J. Optim.

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Abstract. Given a compact parameter set Y ⊂ R p , we consider polynomial optimization problems (Py) on R n whose description depends on the parameter y ∈ Y. We assume that one can compute all moments of some probability measure ϕ on Y, absolutely continuous with respect to the Lebesgue measure (e.g. Y is a box or a simplex and ϕ is uniformly distributed). We then provide a hierarchy of semidefinite relaxations whose associated sequence of optimal solutions converges to the moment vector of a probability measure that encodes all information about all global optimal solutions x * (y) of Py, as y ∈ Y. In particular, one may approximate as closely as desired any polynomial functional of the optimal solutions, like e.g. their ϕ-mean. In addition, using this knowledge on moments, the measurable function y → x * k (y) of the k-th coordinate of optimal solutions, can be estimated, e.g. by maximum entropy methods. Also, for a boolean variable x k , one may approximate as closely as desired its persistency ϕ({y : x * k (y) = 1}, i.e. the probability that in an optimal solution x * (y), the coordinate x * k (y) takes the value 1. At last but not least, from an optimal solution of the dual semidefinite relaxations, one provides a sequence of polynomial (resp. piecewise polynomial) lower approximations with L 1 (ϕ) (resp. almost uniform) convergence to the optimal value function. IntroductionRoughly speaking, given a set parameters Y and an optimization problem whose description depends on y ∈ Y (call it P y ), parametric optimization is concerned with the behavior and properties of the optimal value as well as primal (and possibly dual) optimal solutions of P y , when y varies in Y. This a quite challenging problem and in general one may obtain information locally around some nominal value y 0 of the parameter. There is a vast and rich literature on the topic and for a detailed treatment, the interested reader is referred to e.g. Bonnans and Shapiro [4] and the many references therein. Sometimes, in the context of optimization with data uncertainty, some probability distribution ϕ on the parameter set Y is available and in this context one is also interested in e.g. the distribution of the optimal value, optimal solutions, all viewed as random variables. In particular, for discrete optimization problems where cost coefficients are random variables with joint distribution ϕ, some bounds on the expected optimal value have been obtained. More recently Natarajan et al. [17] extended the earlier work in [3] to even 1991 Mathematics Subject Classification. 65 D15, 65 K05, 46 N10, 90 C22.

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Persistence in discrete optimization under data uncertainty

Cited by 58 publications

References 22 publications

A semidefinite optimization approach to the steady-state analysis of queueing systems

A semidefinite optimization approach to the steady-state analysis of queueing systems

Robustness to Dependency in Portfolio Optimization Using Overlapping Marginals

A “Joint+Marginal” Approach to Parametric Polynomial Optimization

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