Existence and Optimality Conditions for Risk-Averse PDE-Constrained Optimization

Kouri, Drew Philip; Surowiec, Thomas M.

doi:10.1137/16M1086613

Title: Existence and Optimality Conditions for Risk-Averse PDE-Constrained Optimization

Journal Article · Tue Jun 05 00:00:00 EDT 2018 · SIAM/ASA Journal on Uncertainty Quantification

DOI:https://doi.org/10.1137/16M1086613· OSTI ID:1441483

Kouri, Drew Philip ^[1]; Surowiec, Thomas M. ^[2]

Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Philipps-Universitat Marburg (Germany). FB12 Mathematik und Informatik

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.

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Research Organization:: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA); Defense Advanced Research Projects Agency (DARPA)

Grant/Contract Number:: AC04-94AL85000; NA0003525

OSTI ID:: 1441483

Report Number(s):: SAND-2018-5345J; 663230

Journal Information:: SIAM/ASA Journal on Uncertainty Quantification, Vol. 6, Issue 2; ISSN 2166-2525

Publisher:: SIAMCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 28 works

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References (32)

Spectral measures of risk: A coherent representation of subjective risk aversion Acerbi, Carlo Journal of Banking & Finance, Vol. 26, Issue 7 https://doi.org/10.1016/S0378-4266(02)00281-9	journal	July 2002
Multilevel Monte Carlo Analysis for Optimal Control of Elliptic PDEs with Random Coefficients Ali, Ahmad Ahmad; Ullmann, Elisabeth; Hinze, Michael SIAM/ASA Journal on Uncertainty Quantification, Vol. 5, Issue 1 https://doi.org/10.1137/16M109870X	journal	January 2017
Coherent Measures of Risk Artzner, Philippe; Delbaen, Freddy; Eber, Jean-Marc Mathematical Finance, Vol. 9, Issue 3 https://doi.org/10.1111/1467-9965.00068	journal	July 1999
Expected Utility, Penalty Functions, and Duality in Stochastic Nonlinear Programming Ben-Tal, Aharon; Teboulle, Marc Management Science, Vol. 32, Issue 11 https://doi.org/10.1287/mnsc.32.11.1445	journal	November 1986
An Old-New Concept of Convex risk Measures: the Optimized Certainty Equivalent Ben-Tal, Aharon; Teboulle, Marc Mathematical Finance, Vol. 17, Issue 3 https://doi.org/10.1111/j.1467-9965.2007.00311.x	journal	July 2007
Low-rank solvers for unsteady Stokes–Brinkman optimal control problem with random data Benner, Peter; Dolgov, Sergey; Onwunta, Akwum Computer Methods in Applied Mechanics and Engineering, Vol. 304 https://doi.org/10.1016/j.cma.2016.02.004	journal	June 2016
On the treatment of distributed uncertainties in PDE-constrained optimization Borzì, A.; Schulz, V.; Schillings, C. GAMM-Mitteilungen, Vol. 33, Issue 2 https://doi.org/10.1002/gamm.201010017	journal	October 2010
Multigrid Methods and Sparse-Grid Collocation Techniques for Parabolic Optimal Control Problems with Random Coefficients Borzì, A.; von Winckel, G. SIAM Journal on Scientific Computing, Vol. 31, Issue 3 https://doi.org/10.1137/070711311	journal	January 2009
A POD framework to determine robust controls in PDE optimization Borzì, A.; von Winckel, G. Computing and Visualization in Science, Vol. 14, Issue 3 https://doi.org/10.1007/s00791-011-0165-5	journal	March 2011
Weighted Reduced Basis Method for Stochastic Optimal Control Problems with Elliptic PDE Constraint Chen, Peng; Quarteroni, Alfio SIAM/ASA Journal on Uncertainty Quantification, Vol. 2, Issue 1 https://doi.org/10.1137/130940517	journal	January 2014
Stochastic Optimal Robin Boundary Control Problems of Advection-Dominated Elliptic Equations Chen, Peng; Quarteroni, Alfio; Rozza, Gianluigi SIAM Journal on Numerical Analysis, Vol. 51, Issue 5 https://doi.org/10.1137/120884158	journal	January 2013
Convex measures of risk and trading constraints Föllmer, Hans; Schied, Alexander Finance and Stochastics, Vol. 6, Issue 4 https://doi.org/10.1007/s007800200072	journal	October 2002
Constrained Optimization with Low-Rank Tensors and Applications to Parametric Problems with PDEs Garreis, Sebastian; Ulbrich, Michael SIAM Journal on Scientific Computing, Vol. 39, Issue 1 https://doi.org/10.1137/16M1057607	journal	January 2017
On NEMYTSKIJ Operators inLp-Spaces of Abstract Functions Goldberg, H.; Kampowsky, W.; Tröltzsch, F. Mathematische Nachrichten, Vol. 155, Issue 1 https://doi.org/10.1002/mana.19921550110	journal	January 1992
A Multilevel Stochastic Collocation Algorithm for Optimization of PDEs with Uncertain Coefficients Kouri, D. P. SIAM/ASA Journal on Uncertainty Quantification, Vol. 2, Issue 1 https://doi.org/10.1137/130915960	journal	January 2014
A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty Kouri, D. P.; Heinkenschloss, M.; Ridzal, D. SIAM Journal on Scientific Computing, Vol. 35, Issue 4 https://doi.org/10.1137/120892362	journal	January 2013
Inexact Objective Function Evaluations in a Trust-Region Algorithm for PDE-Constrained Optimization under Uncertainty Kouri, D. P.; Heinkenschloss, M.; Ridzal, D. SIAM Journal on Scientific Computing, Vol. 36, Issue 6 https://doi.org/10.1137/140955665	journal	January 2014
Risk-Averse PDE-Constrained Optimization Using the Conditional Value-At-Risk Kouri, D. P.; Surowiec, T. M. SIAM Journal on Optimization, Vol. 26, Issue 1 https://doi.org/10.1137/140954556	journal	January 2016
Model Order Reduction Techniques with a Posteriori Error Control for Nonlinear Robust Optimization Governed by Partial Differential Equations Lass, Oliver; Ulbrich, Stefan SIAM Journal on Scientific Computing, Vol. 39, Issue 5 https://doi.org/10.1137/16M108269X	journal	January 2017
A Sample Approximation Approach for Optimization with Probabilistic Constraints Luedtke, James; Ahmed, Shabbir SIAM Journal on Optimization, Vol. 19, Issue 2 https://doi.org/10.1137/070702928	journal	January 2008
Convex Approximations of Chance Constrained Programs Nemirovski, Arkadi; Shapiro, Alexander SIAM Journal on Optimization, Vol. 17, Issue 4 https://doi.org/10.1137/050622328	journal	January 2007
An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data Nobile, F.; Tempone, R.; Webster, C. G. SIAM Journal on Numerical Analysis, Vol. 46, Issue 5 https://doi.org/10.1137/070680540	journal	January 2008
Sample Average Approximation Method for Chance Constrained Programming: Theory and Applications Pagnoncelli, B. K.; Ahmed, S.; Shapiro, A. Journal of Optimization Theory and Applications, Vol. 142, Issue 2 https://doi.org/10.1007/s10957-009-9523-6	journal	March 2009
Nonsmooth analysis on partially ordered vector spaces. I. Convex case Papageorgiou, Nikolaos Pacific Journal of Mathematics, Vol. 107, Issue 2 https://doi.org/10.2140/pjm.1983.107.403	journal	January 1983
On buffered failure probability in design and optimization of structures Rockafellar, R. T.; Royset, J. O. Reliability Engineering & System Safety, Vol. 95, Issue 5 https://doi.org/10.1016/j.ress.2010.01.001	journal	May 2010
Optimization of conditional value-at-risk Rockafellar, R. Tyrrell; Uryasev, Stanislav The Journal of Risk, Vol. 2, Issue 3 https://doi.org/10.21314/JOR.2000.038	journal	January 2000
Problem Formulations and Treatment of Uncertainties in Aerodynamic Design Schulz, Volker; Schillings, Claudia AIAA Journal, Vol. 47, Issue 3 https://doi.org/10.2514/1.37955	journal	March 2009
Inverse problems: A Bayesian perspective Stuart, A. M. Acta Numerica, Vol. 19 https://doi.org/10.1017/S0962492910000061	journal	May 2010
Stochastic Collocation for Optimal Control Problems with Stochastic PDE Constraints Tiesler, Hanne; Kirby, Robert M.; Xiu, Dongbin SIAM Journal on Control and Optimization, Vol. 50, Issue 5 https://doi.org/10.1137/110835438	journal	January 2012
Derivatives of probability functions and integrals over sets given by inequalities Uryas'ev, Stanislav Journal of Computational and Applied Mathematics, Vol. 56, Issue 1-2 https://doi.org/10.1016/0377-0427(94)90388-3	journal	December 1994
Derivatives of probability functions and some applications Uryasev, Stanislav Annals of Operations Research, Vol. 56, Issue 1 https://doi.org/10.1007/BF02031712	journal	December 1995
(Sub-)Gradient Formulae for Probability Functions of Random Inequality Systems under Gaussian Distribution van Ackooij, Wim; Henrion, R. SIAM/ASA Journal on Uncertainty Quantification, Vol. 5, Issue 1 https://doi.org/10.1137/16M1061308	journal	January 2017

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Spectral risk measures: the risk quadrangle and optimal approximation Kouri, Drew P. Mathematical Programming, Vol. 174, Issue 1-2 https://doi.org/10.1007/s10107-018-1267-3	journal	May 2018

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97 MATHEMATICS AND COMPUTING
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Title: Existence and Optimality Conditions for Risk-Averse PDE-Constrained Optimization

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