A New Scenario-tree Generation Approach for Multistage Stochastic Programming Problems Based on a Demerit Criterion PDF Download
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Author: Julien Keutchayan
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Languages : en
Pages : 25
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Author: Julien Keutchayan
Publisher:
ISBN:
Category :
Languages : en
Pages : 25
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Author: Debora Mahlke
Publisher: Springer Science & Business Media
ISBN: 3834898295
Category : Mathematics
Languages : en
Pages : 194
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Motivated by practical optimization problems occurring in energy systems with regenerative energy supply, Debora Mahlke formulates and analyzes multistage stochastic mixed-integer models. For their solution, the author proposes a novel decomposition approach which relies on the concept of splitting the underlying scenario tree into subtrees. Based on the formulated models from energy production, the algorithm is computationally investigated and the numerical results are discussed.
Author:
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Languages : en
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Modern financial portfolio management problems as well as asset/liability problems use stochastic optimization to allocate financial assets. To implement and solve such a stochastic optimization based portfolio allocation problem, we require scenario trees for the description of the future market evolutions of every random variable present in the model. This thesis proposes a general algorithm to construct scenario trees for underlying assets as well as options on these assets. The algorithm is based on the simulation of GARCH processes and on a Wasserstein distance minimization for the reduction of the number of scenarios. Several processes are analyzed, and empirical results on the DAX 100 and on European Put and Call options on this index are presented.
Author: M. A. H. Dempster
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Languages : en
Pages : 49
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In solving a scenario-based dynamic (multistage) stochastic programme, scenario generation plays a critical role as it forms the input specification to the optimization process. Computational bottlenecks in this process place a limit on the number of scenarios employable in approximating the probability distribution of the paths of the underlying uncertainty. Traditional scenario generation approaches have been to find a sampling method that best approximates the path distribution in terms of some probability metrics such as the minimization of moment deviations or (Monge-Kantotrovich-)Wasserstein distance. Here we present a Wasserstein-based heuristic for discretization of a continuous state path distribution. The paper compares this heuristic to the existing methods in the literature (Monte Carlo sampling, moment matching, Latin Hypercube sampling, scenario reduction, sequential clustering) in terms of their effectiveness in suppressing sampling error when used to generate the scenario tree of a dynamic stochastic programme.We perform an extensive investigation of the impact of scenario generation techniques on the in- and out-of-sample stability of a simplified version of a four-period asset liability management problem employed in practice. A series of out-of-sample tests are carried out to evaluate the effect of possible discretization biases. We also attempt to provide a motivation for the popular utilization of left-heavy scenario trees (i.e. with more early than later period branching) based on the Wasserstein distance criterion. Empirical results show that all methods outperform normal MC sampling. However when evaluated against each other all these methods perform essentially equally well, with second-order moment matching showing only marginal improvements in terms of in-sample stability and out-of-sample performance. The out-of-sample results highlight the under-estimation of portfolio risk which results from insufficient scenario samples. This discretization bias induces overly aggressive portfolio balance recommendations which can impair the performance of the model in real world applications. Thus in future research this issue needs to be carefully addressed, see e.g.
Author: Christian Küchler
Publisher: Springer Science & Business Media
ISBN: 3834893994
Category : Mathematics
Languages : en
Pages : 178
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Christian Küchler studies various aspects of the stability of stochastic optimization problems as well as approximation and decomposition methods in stochastic programming. In particular, the author presents an extension of the Nested Benders decomposition algorithm related to the concept of recombining scenario trees.
Author: René Henrion
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Languages : en
Pages :
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Scenarios are indispensable ingredients for the numerical solution of stochastic programs. Earlier approaches to optimal scenario generation and reduction are based on stability arguments involving distances of probability measures. In this paper we review those ideas and suggest to make use of stability estimates based only on problem specific data. For linear two-stage stochastic programs we show that the problem-based approach to optimal scenario generation can be reformulated as best approximation problem for the expected recourse function which in turn can be rewritten as a generalized semi-infinite program. We show that the latter is convex if either right-hand sides or costs are random and can be transformed into a semi-infinite program in a number of cases. We also consider problem-based optimal scenario reduction for two-stage models and optimal scenario generation for chance constrained programs. Finally, we discuss problem-based scenario generation for the classical newsvendor problem.
Author: Kjetil Høyland
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Languages : en
Pages :
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Author: Georg Schildbach
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Languages : en
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Author:
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Category :
Languages : en
Pages : 24
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We study the use of sparse grids in the scenario generation (or discretization) problem in stochastic programming problems where the uncertainty is modeled using a continuous multivariate distribution. We show that, under a regularity assumption on the random function involved, the sequence of optimal objective function values of the sparse grid approximations converges to the true optimal objective function values as the number of scenarios increases. The rate of convergence is also established. We treat separately the special case when the underlying distribution is an affine transform of a product of univariate distributions, and show how the sparse grid method can be adapted to the distribution by the use of quadrature formulas tailored to the distribution. We numerically compare the performance of the sparse grid method using different quadrature rules with classic quasi-Monte Carlo (QMC) methods, optimal rank-one lattice rules, and Monte Carlo (MC) scenario generation, using a series of utility maximization problems with up to 160 random variables. The results show that the sparse grid method is very efficient, especially if the integrand is sufficiently smooth. In such problems the sparse grid scenario generation method is found to need several orders of magnitude fewer scenarios than MC and QMC scenario generation to achieve the same accuracy. As a result, it is indicated that the method scales well with the dimension of the distribution--especially when the underlying distribution is an affine transform of a product of univariate distributions, in which case the method appears scalable to thousands of random variables.
Author: Julien Keutchayan
Publisher:
ISBN:
Category :
Languages : en
Pages : 23
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