Combining Stochastic Programming and Optimal Control to Solve Multistage Stochastic Optimization Problems

2011 ◽  
Author(s):  
Diana Barro ◽  
Elio Canestrelli
Keyword(s):  
Optimal Control ◽  
Stochastic Programming ◽  
Stochastic Optimization ◽  
Optimization Problems ◽  
Multistage Stochastic Optimization

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 Risk-averse optimal control of semilinear elliptic PDEs

2020 ◽  
Vol 26 ◽  
pp. 53 ◽  
Author(s):  
D.P. Kouri ◽  
T.M. Surowiec
Keyword(s):  
Optimal Control ◽  
Stochastic Optimization ◽  
Optimization Problems ◽  
Risk Measures ◽  
Elliptic Pdes ◽  
Infinite Dimensional ◽  
Continuity Properties ◽  
Semilinear Elliptic ◽  
Solution Regularity ◽  
Semilinear Elliptic Pdes

In this paper, we consider the optimal control of semilinear elliptic PDEs with random inputs. These problems are often nonconvex, infinite-dimensional stochastic optimization problems for which we employ risk measures to quantify the implicit uncertainty in the objective function. In contrast to previous works in uncertainty quantification and stochastic optimization, we provide a rigorous mathematical analysis demonstrating higher solution regularity (in stochastic state space), continuity and differentiability of the control-to-state map, and existence, regularity and continuity properties of the control-to-adjoint map. Our proofs make use of existing techniques from PDE-constrained optimization as well as concepts from the theory of measurable multifunctions. We illustrate our theoretical results with two numerical examples motivated by the optimal doping of semiconductor devices.

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.

Optimization Design of Trusses Based on Covariance Matrix Adaptation Evolution Strategy Algorithm

2012 ◽  
Vol 215-216 ◽  
pp. 133-137
Author(s):  
Guo Shao Su ◽  
Yan Zhang ◽  
Zhen Xing Wu ◽  
Liu Bin Yan
Keyword(s):  
Stochastic Optimization ◽  
Covariance Matrix ◽  
Optimization Problems ◽  
Fitness Function ◽  
Optimization Methods ◽  
Evolution Strategy ◽  
Covariance Matrix Adaptation ◽  
Study Results ◽  
Highly Nonlinear ◽  
Adaptation Evolution

Covariance matrix adaptation evolution strategy algorithm (CMA-ES) is a newly evolution algorithm. It has become a powerful tool for solving highly nonlinear multi-peak optimization problems. In many real-world optimization problems, the location of multiple optima is often required in a search space. In order to evaluate the solution, thousands of fitness function evaluations are involved that is a time consuming or expensive processes. Therefore, conventional stochastic optimization methods meet a special challenge for a very large number of problem function evaluations. Aiming to overcome the shortcoming of stochastic optimization methods in the high calculation cost, a truss optimal method based on CMA-ES algorithm is proposed and applied to solve the section and shape optimization problems of trusses. The study results show that the method is feasible and has the advantages of high accuracy, high efficiency and easy implementation.

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