Consistency and Properties of Large Deviations of Empirical Estimates in Stochastic Optimization Problems for Homogeneous Random Fields under Nonhomogeneous and Homogeneous Observations

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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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.

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