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=== Design Optimization via the Continuous Stochastic Gradient Method ===
=== Design Optimization via the Continuous Stochastic Gradient Method ===
=== Design Optimization via the Continuous Stochastic Gradient Method ===


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We consider large-scale optimization problems with objective functions or constraints depending on random or distributed parameters. Important examples are problems in which expected properties are minimized or problems subject to chance constraints. Here, standard deterministic optimization approaches rely on a discretization of the appearing integrals. However, the underlying quadrature rule can introduce artificial local minima, resulting in an overall poor performance of the optimizer. Since circumventing this effect severely increases the computational cost, stochastic gradient-type methods have gained popularity over the years, most recently also for the solution of topology optimization problems. However, standard schemes from the literature are typically limited to expected loss functions and still require many iterations. This implies that a lot of expensive state problems have to be solved. Thus, we present the continuous stochastic gradient method (CSG), which provides an efficient hybrid approach without these limitations. In CSG, samples calculated in previous iterations are collected in an optimal linear combination to obtain an approximation to the full gradient and objective function value. It can be shown that the approximation error for both quantities vanishes during the optimization process. Therefore, CSG inherits many convergence properties known from full gradient methods, like convergence for constant step sizes. Moreover, the approximation idea in CSG is not limited to simple gradient schemes but can be combined with more elaborated sequential programming techniques. As an example, the combination with the well-known MMA scheme is shown. After presenting the main theoretical properties of CSG, its efficiency is demonstrated for applications from elasticity and/or nano-particle optics.
We consider large-scale optimization problems with objective functions or constraints depending on random or distributed parameters. Important examples are problems in which expected properties are minimized or problems subject to chance constraints. Here, standard deterministic optimization approaches rely on a discretization of the appearing integrals. However, the underlying quadrature rule can introduce artificial local minima, resulting in an overall poor performance of the optimizer. Since circumventing this effect severely increases the computational cost, stochastic gradient-type methods have gained popularity over the years, most recently also for the solution of topology optimization problems. However, standard schemes from the literature are typically limited to expected loss functions and still require many iterations. This implies that a lot of expensive state problems have to be solved. Thus, we present the continuous stochastic gradient method (CSG), which provides an efficient hybrid approach without these limitations. In CSG, samples calculated in previous iterations are collected in an optimal linear combination to obtain an approximation to the full gradient and objective function value. It can be shown that the approximation error for both quantities vanishes during the optimization process. Therefore, CSG inherits many convergence properties known from full gradient methods, like convergence for constant step sizes. Moreover, the approximation idea in CSG is not limited to simple gradient schemes but can be combined with more elaborated sequential programming techniques. As an example, the combination with the well-known MMA scheme is shown. After presenting the main theoretical properties of CSG, its efficiency is demonstrated for applications from elasticity and/or nano-particle optics.


[https://bit.ly/3XNICTm Livestream access] (MS Teams) |
Recorded talk at [https://youtu.be/bArUxAlSlXc YouTube]  


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[https://owncloud.cesnet.cz/index.php/s/STvHLGLM0rnSV3L ICS file]
[https://bit.ly/3XNICTm Livestream access] (MS Teams) | [https://owncloud.cesnet.cz/index.php/s/JyP4F5gWFGgCdm3 ICS file]
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This presentation was supported by the Ministry of Education, Youth, and Sports of Czechia through the project ID [https://starfos.tacr.cz/en/projekty/8J24DE005 8J24DE005] (Free-material optimization for manufacturable modular structures) and CZ.02.01.01/00/22_008/0004590 ([https://roboprox.eu ROBOPROX]).

Latest revision as of 13:50, 1 October 2024

Design Optimization via the Continuous Stochastic Gradient Method

Michael Stingl, Lukas Pflug, and Andrian Uihlein

Chair of Applied Mathematics (Continuous Optimization), Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg

1 October 2024, 10:00-11:00 CET, Room B-168 @ Thákurova 7, 166 29 Prague 6

Abstract:

We consider large-scale optimization problems with objective functions or constraints depending on random or distributed parameters. Important examples are problems in which expected properties are minimized or problems subject to chance constraints. Here, standard deterministic optimization approaches rely on a discretization of the appearing integrals. However, the underlying quadrature rule can introduce artificial local minima, resulting in an overall poor performance of the optimizer. Since circumventing this effect severely increases the computational cost, stochastic gradient-type methods have gained popularity over the years, most recently also for the solution of topology optimization problems. However, standard schemes from the literature are typically limited to expected loss functions and still require many iterations. This implies that a lot of expensive state problems have to be solved. Thus, we present the continuous stochastic gradient method (CSG), which provides an efficient hybrid approach without these limitations. In CSG, samples calculated in previous iterations are collected in an optimal linear combination to obtain an approximation to the full gradient and objective function value. It can be shown that the approximation error for both quantities vanishes during the optimization process. Therefore, CSG inherits many convergence properties known from full gradient methods, like convergence for constant step sizes. Moreover, the approximation idea in CSG is not limited to simple gradient schemes but can be combined with more elaborated sequential programming techniques. As an example, the combination with the well-known MMA scheme is shown. After presenting the main theoretical properties of CSG, its efficiency is demonstrated for applications from elasticity and/or nano-particle optics.

Recorded talk at YouTube


This presentation was supported by the Ministry of Education, Youth, and Sports of Czechia through the project ID 8J24DE005 (Free-material optimization for manufacturable modular structures) and CZ.02.01.01/00/22_008/0004590 (ROBOPROX).