Existence and Optimality Conditions for Risk-Averse PDE-Constrained Optimization
Abstract
Uncertainty is ubiquitous in virtually all engineering applications, and, for such problems, it is inadequate to simulate the underlying physics without quantifying the uncertainty in unknown or random inputs, boundary and initial conditions, and modeling assumptions. Here in this paper, we introduce a general framework for analyzing risk-averse optimization problems constrained by partial differential equations (PDEs). In particular, we postulate conditions on the random variable objective function as well as the PDE solution that guarantee existence of minimizers. Furthermore, we derive optimality conditions and apply our results to the control of an environmental contaminant. Lastly, we introduce a new risk measure, called the conditional entropic risk, that fuses desirable properties from both the conditional value-at-risk and the entropic risk measures.
- Authors:
-
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- Philipps-Universitat Marburg (Germany). FB12 Mathematik und Informatik
- Publication Date:
- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA); Defense Advanced Research Projects Agency (DARPA)
- OSTI Identifier:
- 1441483
- Report Number(s):
- SAND-2018-5345J
Journal ID: ISSN 2166-2525; 663230
- Grant/Contract Number:
- AC04-94AL85000; NA0003525
- Resource Type:
- Accepted Manuscript
- Journal Name:
- SIAM/ASA Journal on Uncertainty Quantification
- Additional Journal Information:
- Journal Volume: 6; Journal Issue: 2; Journal ID: ISSN 2166-2525
- Publisher:
- SIAM
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; risk-averse; PDE-constrained optimization; risk measures; uncertainty quantification; stochastic optimization
Citation Formats
Kouri, Drew Philip, and Surowiec, Thomas M. Existence and Optimality Conditions for Risk-Averse PDE-Constrained Optimization. United States: N. p., 2018.
Web. doi:10.1137/16M1086613.
Kouri, Drew Philip, & Surowiec, Thomas M. Existence and Optimality Conditions for Risk-Averse PDE-Constrained Optimization. United States. https://doi.org/10.1137/16M1086613
Kouri, Drew Philip, and Surowiec, Thomas M. Tue .
"Existence and Optimality Conditions for Risk-Averse PDE-Constrained Optimization". United States. https://doi.org/10.1137/16M1086613. https://www.osti.gov/servlets/purl/1441483.
@article{osti_1441483,
title = {Existence and Optimality Conditions for Risk-Averse PDE-Constrained Optimization},
author = {Kouri, Drew Philip and Surowiec, Thomas M.},
abstractNote = {Uncertainty is ubiquitous in virtually all engineering applications, and, for such problems, it is inadequate to simulate the underlying physics without quantifying the uncertainty in unknown or random inputs, boundary and initial conditions, and modeling assumptions. Here in this paper, we introduce a general framework for analyzing risk-averse optimization problems constrained by partial differential equations (PDEs). In particular, we postulate conditions on the random variable objective function as well as the PDE solution that guarantee existence of minimizers. Furthermore, we derive optimality conditions and apply our results to the control of an environmental contaminant. Lastly, we introduce a new risk measure, called the conditional entropic risk, that fuses desirable properties from both the conditional value-at-risk and the entropic risk measures.},
doi = {10.1137/16M1086613},
journal = {SIAM/ASA Journal on Uncertainty Quantification},
number = 2,
volume = 6,
place = {United States},
year = {Tue Jun 05 00:00:00 EDT 2018},
month = {Tue Jun 05 00:00:00 EDT 2018}
}
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