Time-consistent approximations of risk-averse multistage stochastic optimization problems

2014 ◽  
Vol 153 (2) ◽  
pp. 459-493 ◽  
Author(s):  
Tsvetan Asamov ◽  
Andrzej Ruszczyński
Keyword(s):  
Stochastic Optimization ◽  
Optimization Problems ◽  
Risk Averse ◽  
Multistage Stochastic Optimization ◽  
Consistent Approximations

scholarly journals Mixed Spatial and Temporal Decompositions for Large-Scale Multistage Stochastic Optimization Problems

2020 ◽  
Vol 186 (3) ◽  
pp. 985-1005
Author(s):  
Pierre Carpentier ◽  
Jean-Philippe Chancelier ◽  
Michel De Lara ◽  
François Pacaud
Keyword(s):  
Stochastic Optimization ◽  
Large Scale ◽  
Optimization Problems ◽  
Multistage Stochastic Optimization

scholarly journals Single-stage gradient-based stellarator coil design: stochastic optimization

2021 ◽  
Author(s):  
Florian Wechsung ◽  
Andrew Giuliani ◽  
M. Landreman ◽  
Antoine J Cerfon ◽  
Georg Stadler
Keyword(s):  
Stochastic Optimization ◽  
Value At Risk ◽  
Optimization Problems ◽  
Numerical Study ◽  
Single Stage ◽  
Conditional Value At Risk ◽  
Risk Averse ◽  
Coil Design ◽  
Risk Neutral ◽  
Gradient Based

Abstract We extend the single-stage stellarator coil design approach for quasi-symmetry on axis from [Giuliani et al, 2020] to additionally take into account coil manufacturing errors. By modeling coil errors independently from the coil discretization, we have the flexibility to consider realistic forms of coil errors. The corresponding stochastic optimization problems are formulated using a risk-neutral approach and risk-averse approaches. We present an efficient, gradient-based descent algorithm which relies on analytical derivatives to solve these problems. In a comprehensive numerical study, we compare the coil designs resulting from deterministic and risk-neutral stochastic optimization and find that the risk-neutral formulation results in more robust configurations and reduces the number of local minima of the optimization problem. We also compare deterministic and risk-neutral approaches in terms of quasi-symmetry on and away from the magnetic axis, and in terms of the confinement of particles released close to the axis. Finally, we show that for the optimization problems we consider, a risk-averse objective using the Conditional Value-at-Risk leads to results which are similar to the risk-neutral objective.

scholarly journals Martingale characterizations of risk-averse stochastic optimization problems

2019 ◽  
Vol 181 (2) ◽  
pp. 377-403 ◽  
Author(s):  
Alois Pichler ◽  
Ruben Schlotter
Keyword(s):  
Stochastic Optimization ◽  
Optimization Problems ◽  
Risk Averse

scholarly journals Multiscale stochastic optimization: modeling aspects and scenario generation

2019 ◽  
Vol 75 (1) ◽  
pp. 1-34 ◽  
Author(s):  
Martin Glanzer ◽  
Georg Ch. Pflug
Keyword(s):  
Stochastic Optimization ◽  
Real World ◽  
Optimization Problems ◽  
Decision Problems ◽  
Scenario Generation ◽  
Stochastic Evolution ◽  
Modeling Framework ◽  
Uncertainty Factors ◽  
Multistage Stochastic Optimization ◽  
Computational Solution

Abstract Real-world multistage stochastic optimization problems are often characterized by the fact that the decision maker may take actions only at specific points in time, even if relevant data can be observed much more frequently. In such a case there are not only multiple decision stages present but also several observation periods between consecutive decisions, where profits/costs occur contingent on the stochastic evolution of some uncertainty factors. We refer to such multistage decision problems with encapsulated multiperiod random costs, as multiscale stochastic optimization problems. In this article, we present a tailor-made modeling framework for such problems, which allows for a computational solution. We first establish new results related to the generation of scenario lattices and then incorporate the multiscale feature by leveraging the theory of stochastic bridge processes. All necessary ingredients to our proposed modeling framework are elaborated explicitly for various popular examples, including both diffusion and jump models.

On the stochastic optimization problems of plastic metal-working processes

1997 ◽  
Vol 84 (3) ◽  
pp. 1109-1112 ◽  
Author(s):  
M. B. Gitman ◽  
P. V. Trusov ◽  
S. A. Fedoseev
Keyword(s):  
Stochastic Optimization ◽  
Optimization Problems ◽  
Metal Working ◽  
Plastic Metal ◽  
Working Processes

scholarly journals On quantitative stability in infinite-dimensional optimization under uncertainty

2021 ◽  
Author(s):  
M. Hoffhues ◽  
W. Römisch ◽  
T. M. Surowiec
Keyword(s):  
Stochastic Optimization ◽  
Constrained Optimization ◽  
Optimization Problems ◽  
Optimization Under Uncertainty ◽  
Optimal Solutions ◽  
Probability Metrics ◽  
Pde Constrained Optimization ◽  
Quantitative Stability ◽  
Infinite Dimensional ◽  
Optimal Value

AbstractThe vast majority of stochastic optimization problems require the approximation of the underlying probability measure, e.g., by sampling or using observations. It is therefore crucial to understand the dependence of the optimal value and optimal solutions on these approximations as the sample size increases or more data becomes available. Due to the weak convergence properties of sequences of probability measures, there is no guarantee that these quantities will exhibit favorable asymptotic properties. We consider a class of infinite-dimensional stochastic optimization problems inspired by recent work on PDE-constrained optimization as well as functional data analysis. For this class of problems, we provide both qualitative and quantitative stability results on the optimal value and optimal solutions. In both cases, we make use of the method of probability metrics. The optimal values are shown to be Lipschitz continuous with respect to a minimal information metric and consequently, under further regularity assumptions, with respect to certain Fortet-Mourier and Wasserstein metrics. We prove that even in the most favorable setting, the solutions are at best Hölder continuous with respect to changes in the underlying measure. The theoretical results are tested in the context of Monte Carlo approximation for a numerical example involving PDE-constrained optimization under uncertainty.

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