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24
Scenario Reduction in Stochastic Programming: An Approach Using Probability Metrics
 MATH. PROGRAM., SER. A
, 2003
"... Given a convex stochastic programming problem with a discrete initial probability distribution, the problem of optimal scenario reduction is stated as follows: Determine a scenario subset of prescribed cardinality and a probability measure based on this set that is the closest to the initial distr ..."
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Cited by 53 (14 self)
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Given a convex stochastic programming problem with a discrete initial probability distribution, the problem of optimal scenario reduction is stated as follows: Determine a scenario subset of prescribed cardinality and a probability measure based on this set that is the closest to the initial distribution in terms of a natural (or canonical) probability metric. Arguments from stability analysis indicate that FortetMourier type probability metrics may serve as such canonical metrics. Efficient algorithms are developed that determine optimal reduced measures approximately. Numerical experience is reported for reductions of electrical load scenario trees for power management under uncertainty. For instance, it turns out that after 50 % reduction of the scenario tree the optimal reduced tree still has about 90 % relative accuracy.
Scenario Reduction Algorithms in Stochastic Programming
 Computational Optimization and Applications
, 2003
"... We consider convex stochastic programs with an (approximate) initial probability distribution P having nite support supp P , i.e., nitely many scenarios. Such stochastic programs behave stable with respect to perturbations of P measured in terms of a FortetMourier probability metric. The problem ..."
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Cited by 44 (13 self)
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We consider convex stochastic programs with an (approximate) initial probability distribution P having nite support supp P , i.e., nitely many scenarios. Such stochastic programs behave stable with respect to perturbations of P measured in terms of a FortetMourier probability metric. The problem of optimal scenario reduction consists in determining a probability measure which is supported by a subset of supp P of prescribed cardinality and is closest to P in terms of such a probability metric. Two new versions of forward and backward type algorithms are presented for computing such optimally reduced probability measures approximately. Compared to earlier versions, the computational performance (accuracy, running time) of the new algorithms is considerably improved. Numerical experience is reported for dierent instances of scenario trees with computable optimal lower bounds. The test examples also include a ternary scenario tree representing the weekly electrical load process in a power management model.
Polyhedral risk measures in stochastic programming
 SIAM JOURNAL ON OPTIMIZATION
, 2005
"... We consider stochastic programs with risk measures in the objective and study stability properties as well as decomposition structures. Thereby we place emphasis on dynamic models, i.e., multistage stochastic programs with multiperiod risk measures. In this context, we define the class of polyhedra ..."
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Cited by 36 (10 self)
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We consider stochastic programs with risk measures in the objective and study stability properties as well as decomposition structures. Thereby we place emphasis on dynamic models, i.e., multistage stochastic programs with multiperiod risk measures. In this context, we define the class of polyhedral risk measures such that stochastic programs with risk measures taken from this class have favorable properties. Polyhedral risk measures are defined as optimal values of certain linear stochastic programs where the arguments of the risk measure appear on the righthand side of the dynamic constraints. Dual representations for polyhedral risk measures are derived and used to deduce criteria for convexity and coherence. As examples of polyhedral risk measures we propose multiperiod extensions of the ConditionalValueatRisk.
Scenario tree modeling for multistage stochastic programs
 Preprint 296, DFG Research Center Matheon, Berlin 2005, and submitted to Mathematical Programming. Contents Heitsch, H., Römisch, W.: Stability and scenario trees for multistage stochastic programs, Preprint 324, DFG Research Center Matheon, Berlin 2006,
"... programs ..."
Optimization with stochastic dominance constraints
 SIAM Journal on Optimization
"... We consider the problem of constructing a portfolio of finitely many assets whose returns are described by a discrete joint distribution. We propose a new portfolio optimization model involving stochastic dominance constraints on the portfolio return. We develop optimality and duality theory for the ..."
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Cited by 32 (5 self)
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We consider the problem of constructing a portfolio of finitely many assets whose returns are described by a discrete joint distribution. We propose a new portfolio optimization model involving stochastic dominance constraints on the portfolio return. We develop optimality and duality theory for these models. We construct equivalent optimization models with utility functions. Numerical illustration is provided.
STABILITY OF MULTISTAGE STOCHASTIC PROGRAMS
 SIAM J. OPTIM.
, 2006
"... Quantitative stability of linear multistage stochastic programs is studied. It is shown that the infima of such programs behave (locally) Lipschitz continuous with respect to the sum of an L_rdistance and of a distance measure for the filtrations of the original and approximate stochastic (input) ..."
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Cited by 31 (8 self)
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Quantitative stability of linear multistage stochastic programs is studied. It is shown that the infima of such programs behave (locally) Lipschitz continuous with respect to the sum of an L_rdistance and of a distance measure for the filtrations of the original and approximate stochastic (input) processes. Various issues of the result are discussed and an illustrative example is given. Consequences for the reduction of scenario trees are also discussed.
Power management in a hydrothermal system under uncertainty by Lagrangian relaxation
 IMA VOLUMES IN MATHEMATICS AND ITS APPLICATIONS VOL. 128, SPRINGERVERLAG
, 2002
"... We present a dynamic multistage stochastic programming model for the costoptimal generation of electric power in a hydrothermal system under uncertainty in load, inflow to reservoirs and prices for fuel and delivery contracts. The stochastic load process is approximated by a scenario tree obtained ..."
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Cited by 16 (12 self)
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We present a dynamic multistage stochastic programming model for the costoptimal generation of electric power in a hydrothermal system under uncertainty in load, inflow to reservoirs and prices for fuel and delivery contracts. The stochastic load process is approximated by a scenario tree obtained by adapting a SARIMA model to historical data, using empirical means and variances of simulated scenarios to construct an initial tree, and reducing it by a scenario deletion procedure based on a suitable probability distance. Our model involves many mixedinteger variables and individual power unit constraints, but relatively few coupling constraints. Hence we employstochastic Lagrangian relaxation that assigns stochastic multipliers to the coupling constraints. Solving the Lagrangian dual by a proximal bundle method leads to successive decomposition into single thermal and hydro unit subproblems that are solved by dynamic programming and a specialized descent algorithm, respectively. The optimal stochastic multipliers are used in Lagrangian heuristics to construct approximately optimal first stage decisions. Numerical results are presented for realistic data from a German power utility, with a time horizon of one week and scenario numbers ranging from 5 to 100. The corresponding optimization problems have up to 200,000 binary and 350,000 continuous variables, and more than 500,000 constraints.
A note on scenario reduction for twostage stochastic programs
 OPERATIONS RESEARCH LETTERS
, 2007
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Ch.: Financial scenario generation for stochastic multistage decision processes as facility location problem
 Annals of Operations Research
"... Abstract The quality of multistage stochastic optimization models as they appear in asset liability management, energy planning, transportation, supply chain management, and other applications depends heavily on the quality of the underlying scenario model, describing the uncertain processes influe ..."
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Cited by 16 (1 self)
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Abstract The quality of multistage stochastic optimization models as they appear in asset liability management, energy planning, transportation, supply chain management, and other applications depends heavily on the quality of the underlying scenario model, describing the uncertain processes influencing the profit/cost function, such as asset prices and liabilities, the energy demand process, demand for transportation, and the like. A common approach to generate scenarios is based on estimating an unknown distribution and matching its moments with moments of a discrete scenario model. This paper demonstrates that the problem of finding valuable scenario approximations can be viewed as the problem of optimally approximating a given distribution with some distance function. We show that for Lipschitz continuous cost/profit functions it is best to employ the Wasserstein distance. The resulting optimization problem can be viewed as a multidimensional facility location problem, for which at least good heuristic algorithms exist. For multistage problems, a scenario tree is constructed as a nested facility location problem. Numerical convergence results for financial meanrisk portfolio selection conclude the paper. Keywords Stochastic programming. Multistage financial scenario generation 1