We analyze a potentially risk-averse convex stochastic optimization problem, where the control is deterministic and the state is a Banach-valued essentially bounded random variable. We obtain strong forms of necessary and sufficient optimality conditions for problems subject to equality and conical constraints. We propose a Moreau-Yosida regularization for the conical constraint and show consistency of the optimality conditions for the regularized problem as the regularization parameter is taken to infinity.
Keywords: Optimization in Banach spaces, optimality conditions, regularization, convex stochastic optimization in Banach spaces, two-stage stochastic optimization, duality, PDE-constrained optimization under uncertainty
@article{COCV_2022__28_1_A80_0,
author = {Geiersbach, Caroline and Hinterm\"uller, Michael},
title = {Optimality {Conditions} and {Moreau{\textendash}Yosida} {Regularization} for {Almost} {Sure} {State} {Constraints}},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2022070},
mrnumber = {4525175},
zbl = {1509.49015},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2022070/}
}
TY - JOUR AU - Geiersbach, Caroline AU - Hintermüller, Michael TI - Optimality Conditions and Moreau–Yosida Regularization for Almost Sure State Constraints JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2022 VL - 28 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2022070/ DO - 10.1051/cocv/2022070 LA - en ID - COCV_2022__28_1_A80_0 ER -
%0 Journal Article %A Geiersbach, Caroline %A Hintermüller, Michael %T Optimality Conditions and Moreau–Yosida Regularization for Almost Sure State Constraints %J ESAIM: Control, Optimisation and Calculus of Variations %D 2022 %V 28 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2022070/ %R 10.1051/cocv/2022070 %G en %F COCV_2022__28_1_A80_0
Geiersbach, Caroline; Hintermüller, Michael. Optimality Conditions and Moreau–Yosida Regularization for Almost Sure State Constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 80. doi: 10.1051/cocv/2022070
[1] and , Sobolev Spaces, vol. 140. Elsevier (2003). | MR | Zbl
[2] , , and , Mean-variance risk-averse optimal control of systems governed by PDEs with random parameter fields using quadratic approximations. SIAM/ASA J. Uncertainty Quantif. 5 (2017) 1166-1192. | MR | Zbl | DOI
[3] and , Vol. 95 of Nonlinear superposition operators. Cambridge University Press (1990). | MR | Zbl
[4] , , and , Coherent measures of risk. Math. Finance 9 (1999) 203-228. | MR | Zbl | DOI
[5] and , Set-valued analysis. Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA (1990). | MR | Zbl
[6] and , Remarks on Chacon’s biting lemma. Proc. Am. Math. Soc. 107 (1989) 655-663. | MR | Zbl
[7] , et al., Vol. 408 of Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer (2011). | MR | Zbl
[8] , and , Block-diagonal preconditioning for optimal control problems constrained by PDEs with uncertain inputs. SIAM J. Matrix Anal. Appi. 37 (2016) 491-518. | MR | Zbl | DOI
[9] and , Optimal control of linear bottleneck problems. ESAIM: COCV 3 (1998) 235-250. | MR | Zbl | Numdam
[10] and , Perturbation analysis of optimization problems. Springer-Verlag, New York (2013). | MR | Zbl
[11] , Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 24 (1986) 1309-1318. | MR | Zbl | DOI
[12] and , Taylor approximation for chance constrained optimization problems governed by partial differential equations with high-dimensional random parameters. SIAM/ASA J. Uncert. Quantif. 9 (2021) 1381-1410. | MR | Zbl | DOI
[13] , and , A weighted reduced basis method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 51 (2013) 3163-3185. | MR | Zbl | DOI
[14] , , , and , Shape optimization under uncertainty-a stochastic programming perspective. SIAM J. Optim. 19 (2008) 1610-1632. | MR | Zbl | DOI
[15] and , Subregular recourse in nonlinear multistage stochastic optimization. Math. Program. (2021) 1-22. | MR | Zbl
[16] , , , , and , Stochastic optimization models for risk-based reservoir management. Cybern. Syst. Anal. 55 (2019) 55-64. | MR | Zbl | DOI
[17] , and , Properties of chance constraints in infinite dimensions with an application to PDE constrained optimization. Set-Valued Variat. Anal. 26 (2018) 821-841. | MR | Zbl | DOI
[18] and , The canonical model space for law-invariant convex risk measures is . Math. Finance 22 (2012) 585-589. | MR | Zbl | DOI
[19] , and , Risk-neutral PDE-constrained generalized Nash equilibrium problems. Math. Program. (2022) 1-51. | MR | Zbl
[20] and , A stochastic gradient method with mesh refinement for PDE-constrained optimization under uncertainty. SIAM J. Scient. Comput. 42 (2020) A2750-A2772. | MR | Zbl | DOI
[21] and , Optimality conditions for convex stochastic optimization problems in banach spaces with almost sure state constraints. SIAM J. Optim. 4 (2021) 2455-2480. | MR | Zbl | DOI
[22] , , and , Chance constrained optimization of elliptic PDE systems with a smoothing convex approximation. ESAIM: COCV 26 (2020) 70. | MR | Zbl | Numdam
[23] and , Feasible and noninterior path-following in constrained minimization with low multiplier regularity. SIAM J. Control Optim. 45 (2006) 1198-1221. | MR | Zbl | DOI
[24] and , Path-following methods for a class of constrained minimization problems in function space. SIAM J. Optim. 17 (2006) 159-187. | MR | Zbl | DOI
[25] and , Subdifferentials of convex functions. Trudy Moskovskogo Matematicheskogo Obshchestva 26 (1972) 3-73. | MR | Zbl
[26] , and , An approach for robust PDE-constrained optimization with application to shape optimization of electrical engines and of dynamic elastic structures under uncertainty. Optim. Eng. 19 (2018) 697-731. | MR | Zbl | DOI
[27] , , and , A trust-region algorithm with adaptive stochastic collocation for PDE optimization under uncertainty. SIAM J. Sci. Comput. 35 (2013) A1847-A1879. | MR | Zbl | DOI
[28] and , Risk-averse PDE-constrained optimization using the conditional value-at-risk. SIAM J. Optim. 26 (2016) 365-396. | MR | Zbl | DOI
[29] and , Existence and optimality conditions for risk-averse PDE-constrained optimization. SIAM/ASA J. Uncertainty Quant. 6 (2018) 787-815. | MR | Zbl | DOI
[30] , The Lebesgue decomposition for functionals on the vector-function space . Funct. Anal. Appi. 8 (1974) 314-317. | MR | Zbl | DOI
[31] and , Financial planning via multi-stage stochastic optimization. Comput. Oper. Res. 31 (2004) 1-20. | MR | Zbl | DOI
[32] and , Multi-stage stochastic optimization applied to energy planning. Math. Program. 52 (1991) 359-375. | MR | Zbl | DOI
[33] and , Multistage Stochastic Optimization, Springer (2014). | MR | Zbl | DOI
[34] and , Modeling, Measuring and Managing Risk. World Scientific (2007). | MR | Zbl | DOI
[35] , Integrals which are convex functionals. II. Pacific J. Math. 39 (1971) 439-469. | MR | Zbl | DOI
[36] , Convex integral functionals and duality, in Contributions to nonlinear functional analysis. Elsevier (1971), pp. 215-236. | MR | Zbl
[37] , Conjugate duality and optimization. SIAM (1974). | DOI | MR | Zbl
[38] , Integral functionals, normal integrands and measurable selections, in Nonlinear operators and the calculus of variations. Springer (1976), pp. 157-207. | MR | Zbl
[39] and , Engineering decisions under risk averseness. ASCE-ASME J. Risk Uncert. Eng. Syst. A 1 (2015) 04015003. | DOI
[40] and , Stochastic convex programming: Kuhn-Tucker conditions. J. Math. Econ. 2 (1975) 349-370. | MR | Zbl | DOI
[41] and , Stochastic convex programming: basic duality. Pacific J. Math. 62 (1976) 173-195. | MR | Zbl | DOI
[42] and , Stochastic convex programming: relatively complete recourse and induced feasibility. SIAM J. Control Optim. 14 (1976) 574-589. | MR | Zbl | DOI
[43] and , Stochastic convex programming: singular multipliers and extended duality singular multipliers and duality. Pacific J. Math. 62 (1976) 507-522. | MR | Zbl | DOI
[44] , and , Efficient shape optimization for certain and uncertain aerodynamic design. Comput. Fluids 46 (2011) 78-87. | MR | Zbl | DOI
[45] , and , Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia (2009). | MR | Zbl | DOI
[46] , Extensions and decompositions of vector measures. J. London Math. Soc. 2 (1971) 672-676. | MR | Zbl | DOI
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