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82
Robust Dynamic Programming
 Math. Oper. Res
, 2004
"... In this paper we propose a robust formulation for discrete time dynamic programming (DP). The objective of the robust formulation is to systematically mitigate the sensitivity of the DP optimal policy to ambiguity in the underlying transition probabilities. The ambiguity is modeled by associating ..."
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Cited by 40 (1 self)
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In this paper we propose a robust formulation for discrete time dynamic programming (DP). The objective of the robust formulation is to systematically mitigate the sensitivity of the DP optimal policy to ambiguity in the underlying transition probabilities. The ambiguity is modeled by associating a set of conditional measures with each stateaction pair. Consequently, in the robust formulation each policy has a set of measures associated with it. We prove that when this set of measures has a certain "Rectangularity" property all the main results for finite and infinite horizon DP extend to natural robust counterparts. We identify families of sets of conditional measures for which the computational complexity of solving the robust DP is only modestly larger than solving the DP, typically logarithmic in the size of the state space. These families of sets are constructed from the confidence regions associated with density estimation, and therefore, can be chosen to guarantee any desired level of confidence in the robust optimal policy. Moreover, the sets can be easily parameterized from historical data. We contrast the performance of robust and nonrobust DP on small numerical examples.
A selective overview of nonparametric methods in financial econometrics
 Statist. Sci
, 2005
"... Abstract. This paper gives a brief overview of the nonparametric techniques that are useful for financial econometric problems. The problems include estimation and inference for instantaneous returns and volatility functions of timehomogeneous and timedependent diffusion processes, and estimation ..."
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Cited by 35 (8 self)
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Abstract. This paper gives a brief overview of the nonparametric techniques that are useful for financial econometric problems. The problems include estimation and inference for instantaneous returns and volatility functions of timehomogeneous and timedependent diffusion processes, and estimation of transition densities and state price densities. We first briefly describe the problems and then outline the main techniques and main results. Some useful probabilistic aspects of diffusion processes are also briefly summarized to facilitate our presentation and applications.
Ambiguous Chance Constrained Problems And Robust Optimization
 Mathematical Programming
, 2004
"... In this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain. We primarily focus on the special case where the uncertainty set Q of the distributions is of the form Q = {Q : # p (Q, Q 0 ) # #}, where # p denote ..."
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Cited by 35 (1 self)
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In this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain. We primarily focus on the special case where the uncertainty set Q of the distributions is of the form Q = {Q : # p (Q, Q 0 ) # #}, where # p denotes the Prohorov metric. The ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure Q 0 . The main contribution of this paper is to show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with high probability. This result is established using the StrassenDudley Representation Theorem that states that when the distributions of two random variables are close in the Prohorov metric one can construct a coupling of the random variables such that the samples are close with high probability. We also show that the robust sampled problem can be solved e#ciently both in theory and in practice. 1
TWOSTAGE ROBUST NETWORK FLOW AND DESIGN UNDER DEMAND UNCERTAINTY
 FORTHCOMING IN OPERATIONS RESEARCH
, 2004
"... We describe a twostage robust optimization approach for solving network flow and design problems with uncertain demand. In twostage network optimization one defers a subset of the flow decisions until after the realization of the uncertain demand. Availability of such a recourse action allows one ..."
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Cited by 34 (3 self)
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We describe a twostage robust optimization approach for solving network flow and design problems with uncertain demand. In twostage network optimization one defers a subset of the flow decisions until after the realization of the uncertain demand. Availability of such a recourse action allows one to come up with less conservative solutions compared to singlestage optimization. However, this advantage often comes at a price: twostage optimization is, in general, significantly harder than singestage optimization. For network flow and design under demand uncertainty we give a characterization of the firststage robust decisions with an exponential number of constraints and prove that the corresponding separation problem is N Phard even for a network flow problem on a bipartite graph. We show, however, that if the secondstage network topology is totally ordered or an arborescence, then the separation problem is tractable. Unlike singlestage robust optimization under demand uncertainty, twostage robust optimization allows one to control conservatism of the solutions by means of an allowed “budget for demand uncertainty.” Using a budget of uncertainty we provide an upper
Cuts for mixed 01 conic programming
, 2005
"... In this we paper we study techniques for generating valid convex constraints for mixed 01 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 01 linear programs, such as the Gomory cuts, the liftandproject cuts, and cuts from other hierarchies of ti ..."
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Cited by 29 (0 self)
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In this we paper we study techniques for generating valid convex constraints for mixed 01 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 01 linear programs, such as the Gomory cuts, the liftandproject cuts, and cuts from other hierarchies of tighter relaxations, extend in a straightforward manner to mixed 01 conic programs. We also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts. Gomory cuts for mixed 01 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems, e.g. we show that all the subtour elimination inequalities for the traveling salesman problem are rank1 Gomory cuts with respect to a single semidefinite constraint. We also include results from our preliminary computational experiments with these cuts.
A survey of the Slemma
 SIAM Review
"... Abstract. In this survey we review the many faces of the Slemma, a result about the correctness of the Sprocedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as ..."
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Cited by 26 (0 self)
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Abstract. In this survey we review the many faces of the Slemma, a result about the correctness of the Sprocedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as well. These were all active research areas, but as there was little interaction between researchers in these different areas, their results remained mainly isolated. Here we give a unified analysis of the theory by providing three different proofs for the Slemma and revealing hidden connections with various areas of mathematics. We prove some new duality results and present applications from control theory, error estimation, and computational geometry. Key words. Slemma, Sprocedure, control theory, nonconvex theorem of alternatives, numerical range, relaxation theory, semidefinite optimization, generalized convexities
WORSTCASE VALUEATRISK AND ROBUST PORTFOLIO OPTIMIZATION: A CONIC PROGRAMMING APPROACH
, 2001
"... Classical formulations of the portfolio optimization problem, such as meanvariance or ValueatRisk (VaR) approaches, can result in a portfolio extremely sensitive to errors in the data, such as mean and covariance matrix of the returns. In this paper we propose a way to alleviate this problem in a ..."
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Cited by 26 (1 self)
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Classical formulations of the portfolio optimization problem, such as meanvariance or ValueatRisk (VaR) approaches, can result in a portfolio extremely sensitive to errors in the data, such as mean and covariance matrix of the returns. In this paper we propose a way to alleviate this problem in a tractable manner. We assume that the distribution of returns is partially known, in the sense that only bounds on the mean and covariance matrix are available. We define the worstcase ValueatRisk as the largest VaR attainable, given the partial information on the returns ’ distribution. We consider the problem of computing and optimizing the worstcase VaR, and we show that these problems can be cast as semidefinite programs. We extend our approach to various other partial information on the distribution, including uncertainty in factor models, support constraints, and relative entropy information.
Theory and applications of Robust Optimization
, 2007
"... In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most pr ..."
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Cited by 23 (5 self)
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In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most prominent theoretical results of RO over the past decade, we will also present some recent results linking RO to adaptable models for multistage decisionmaking problems. Finally, we will highlight successful applications of RO across a wide spectrum of domains, including, but not limited to, finance, statistics, learning, and engineering.
Disciplined convex programming
 Global Optimization: From Theory to Implementation, Nonconvex Optimization and Its Application Series
, 2006
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