Martingale characterizations of risk-averse stochastic optimization problems

Alois Pichler; Ruben Schlotter

doi:10.1007/s10107-019-01391-2

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

Nuclear Fusion ◽

10.1088/1741-4326/ac45f3 ◽

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.

Download Full-text

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

Mathematical Programming ◽

10.1007/s10107-014-0813-x ◽

2014 ◽

Vol 153 (2) ◽

pp. 459-493 ◽

Cited By ~ 16

Author(s):

Tsvetan Asamov ◽

Andrzej Ruszczyński

Keyword(s):

Stochastic Optimization ◽

Optimization Problems ◽

Risk Averse ◽

Multistage Stochastic Optimization ◽

Consistent Approximations

Download Full-text

On the stochastic optimization problems of plastic metal-working processes

Journal of Mathematical Sciences ◽

10.1007/bf02398421 ◽

1997 ◽

Vol 84 (3) ◽

pp. 1109-1112 ◽

Cited By ~ 1

Author(s):

M. B. Gitman ◽

P. V. Trusov ◽

S. A. Fedoseev

Keyword(s):

Stochastic Optimization ◽

Optimization Problems ◽

Metal Working ◽

Plastic Metal ◽

Working Processes

Download Full-text

On quantitative stability in infinite-dimensional optimization under uncertainty

Optimization Letters ◽

10.1007/s11590-021-01707-2 ◽

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.

Download Full-text

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

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.215-216.133 ◽

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.

Download Full-text