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Title: 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:
 [1];  [2]
  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
  2. 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:
Journal Article: 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. doi:10.1137/16M1086613.
Kouri, Drew Philip, and Surowiec, Thomas M. Tue . "Existence and Optimality Conditions for Risk-Averse PDE-Constrained Optimization". United States. doi:10.1137/16M1086613.
@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}
}

Journal Article:
Free Publicly Available Full Text
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