Chance constrained optimization of elliptic PDE systems with a smoothing convex approximation
ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 70.

In this paper, we consider chance constrained optimization of elliptic partial differential equation (CCPDE) systems with random parameters and constrained state variables. We demonstrate that, under standard assumptions, CCPDE is a convex optimization problem. Since chance constrained optimization problems are generally nonsmooth and difficult to solve directly, we propose a smoothing inner-outer approximation method to generate a sequence of smooth approximate problems for the CCPDE. Thus, the optimal solution of the convex CCPDE is approximable through optimal solutions of the inner-outer approximation problems. A numerical example demonstrates the viability of the proposed approach.

DOI : 10.1051/cocv/2019077
Classification : 46E35, 46B09, 49K30, 90C15, 90C25, 90C30
Mots-clés : Chance constraints, stochastic optimization, elliptic PDEs systems, random parameters, smoothing, inner-outer approximation 
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     author = {Geletu, Abebe and Hoffmann, Armin and Schmidt, Patrick and Li, Pu},
     title = {Chance constrained optimization of elliptic {PDE} systems with a smoothing convex approximation},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {26},
     year = {2020},
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     mrnumber = {4155223},
     zbl = {1451.90105},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2019077/}
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Geletu, Abebe; Hoffmann, Armin; Schmidt, Patrick; Li, Pu. Chance constrained optimization of elliptic PDE systems with a smoothing convex approximation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 70. doi : 10.1051/cocv/2019077. http://archive.numdam.org/articles/10.1051/cocv/2019077/

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The authors would like to express their indebtedness to the Deutsche Forschungsgemeinschaft (DFG) for the financial support under the grants Nr. LI806/13-1 and Nr. LI806/13-2.