Epi-Regularization of Risk Measures
Published Online:16 Aug 2019https://doi.org/10.1287/moor.2019.1013
References
- [1] (2017) Mean-variance risk-averse optimal control of systems governed by PDEs with random parameter fields using quadratic approximations. SIAM/ASA J. Uncertainty Quantification 5(1):1166–1192.Crossref, Google Scholar
- [2] (1999) Coherent measures of risk. Math. Finance 9(3):203–228.Crossref, Google Scholar
- [3] (1984) Variational Convergence for Functions and Operators, Applicable Mathematics Series (Pitman Advanced Publishing Program, Boston).Google Scholar
- [4] (1996) Viscosity solutions of minimization problems. SIAM J. Optim. 6(3):769–806.Crossref, Google Scholar
- [5] (1989) Epigraphical analysis. Annales de l’institut Henri Poincaré (C) Analyse non linéaire S6:73–100.Google Scholar
- [6] (2006) Variational Analysis in Sobolev and BV Spaces, vol. 6 of MPS/SIAM Series on Optimization (SIAM, Philadelphia).Google Scholar
- [7] (2015) Time-varying process control for stringer sheet forming by a deterministic derivative-free optimization approach. Internat. J. Adv. Manufacturing Tech. 80(5):817–828.Crossref, Google Scholar
- [8] (1977) Quelque propriétés des opérateurs angle-bornés et n-cycliquement monotones. Israel J. Math. 26(2):137–150.Crossref, Google Scholar
- [9] (2010) The Baillon-Haddad theorem revisited. J. Convex Anal. 17(3):781–787.Google Scholar
- [10] (1986) Expected utility, penalty functions, and duality in stochastic nonlinear programming. Management Sci. 32(11):1445–1466.Link, Google Scholar
- [11] (2007) An old-new concept of convex risk measures: The optimized certainty equivalent. Math. Finance 17(3):449–476.Crossref, Google Scholar
- [12] (2017) A multi-fidelity framework for investigating the performance of super-cavitating hydrofoils under uncertain flow conditions. 19th AIAA Non-Deterministic Approaches Conf. (American Institute of Aeronautics and Astronautics, Reston, VA).Google Scholar
- [13] (2000) Perturbation Analysis of Optimization Problems (Springer Verlag, New York).Crossref, Google Scholar
- [14] (2006) Numerical Optimization: Theoretical and Practical Aspects, Universitext (Springer-Verlag, Berlin Heidelberg).Google Scholar
- [15] (2011) A POD framework to determine robust controls in PDE optimization. Comput. Vision Sci. 14:91–103.Crossref, Google Scholar
- [16] (2010) On the treatment of distributed uncertainties in PDE constrained optimization. GAMM-Mitt. 33(2):230–246.Crossref, Google Scholar
- [17] (2017) Epi-convergence properties of smoothing by infimal convolution. Set-Valued Variational Anal. 25(1):1–23.Crossref, Google Scholar
- [18] (2014) Weighted reduced basis method for stochastic optimal control problems with elliptic PDE constraint. SIAM/ASA J. Uncertainty Quantification 2(1):364–396.Crossref, Google Scholar
- [19] (2013) Stochastic optimal robin boundary control problems of advection-dominated elliptic equations. SIAM J. Numer. Anal. 51(5):2700–2722.Crossref, Google Scholar
- [20] (1983) Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts (John Wiley & Sons, New York).Google Scholar
- [21] (2000) Trust–Region Methods (SIAM, Philadelphia).Crossref, Google Scholar
- [22] (1993) An Introduction to Γ-Convergence. Progress in Nonlinear Differential Equations and Their Applications (Birkhäuser, Boston).Google Scholar
- [23] (2003) Optimization with stochastic dominance constraints. SIAM J. Optim. 14(2):548–566.Crossref, Google Scholar
- [24] (2012) Uncertainty quantification for subsurface flow problems using coarse-scale models. Graham IG, Hou TY, Lakkis O, Scheichl R, eds. Numerical analysis of multiscale problems, Lecture Notes in Computational Science and Engineering, vol. 83 (Springer, Berlin Heidelberg), 163–202.Crossref, Google Scholar
- [25] (1999) Convex Analysis and Variational Problems, Classics in Applied Mathematics, vol. 28 (SIAM, Philadelphia).Crossref, Google Scholar
- [26] (1999) Real Analysis. Modern Techniques and Their Applications, Pure and Applied Mathematics, 2nd ed. (John Wiley & Sons, New York).Google Scholar
- [27] (2002) Convex measures of risk and trading constraints. Finance Stochastics 6(4):429–447.Crossref, Google Scholar
- [28] (1992) On NEYMTSKIJ operators in Lp-spaces of abstract functions. Mathematische Nachrichten 155(1):127–140.Crossref, Google Scholar
- [29] (1957) Functional analysis and semi-groups. American Mathematical Society Colloquium Publications, vol. 31, rev. ed. (American Mathematical Society, Providence, RI).Google Scholar
- [30] (2009) Moreau-Yosida regularization in state constrained elliptic control problems: Error estimates and parameter adjustment. SIAM J. Numer. Anal. 47(3):1666–1683.Crossref, Google Scholar
- [31] (2006) Feasible and noninterior path-following in constrained minimization with low multiplier regularity. SIAM J. Control Optim. 45(4):1198–1221.Crossref, Google Scholar
- [32] (2006) Path-following methods for a class of constrained minimization problems in function space. SIAM J. Optim. 17(1):159–187.Crossref, Google Scholar
- [33] (2014) A multilevel stochastic collocation algorithm for optimization of PDEs with uncertain coefficients. SIAM/ASA J. Uncertainty Quantification 2(1):55–81.Crossref, Google Scholar
- [34] (2017) A measure approximation for distributionally robust PDE-constrained optimization problems. SIAM J. Numer. Anal. 55(6):3147–3172.Google Scholar
- [35] (2016) Risk-averse PDE-constrained optimization using the conditional value-at-risk. SIAM J. Optim. 26(1):365–396.Google Scholar
- [36] (2017) PDE-constrained optimization under uncertainty. SIAM SIAG/OPT Views News 25(2):1–8.Google Scholar
- [37] (2018) Existence and optimality conditions for risk-averse PDE-constrained optimization. SIAM/ASA J. Uncertainty Quantification 6(2):787–815.Google Scholar
- [38] (2017) ROL: Rapid optimization library. Accessed October 10, 2017, https://trilinos.org/packages/rol.Google Scholar
- [39] (2013) A trust-region algorithm with adaptive stochastic collocation for PDE optimization under uncertainty. SIAM J. Sci. Comput. 35(4):A1847–A1879.Crossref, Google Scholar
- [40] (2014) Inexact objective function evaluations in a trust-region algorithm for PDE-constrained optimization under uncertainty. SIAM J. Sci. Comput. 36(6):A3011–A3029.Crossref, Google Scholar
- [41] (2002) Functional Analysis (John Wiley & Sons, New-York).Google Scholar
- [42] (1971) Optimal Control of Systems Governed by Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, vol. 170 (Springer-Verlag, Berlin Heidelberg).Google Scholar
- [43] (1992) Nonsmooth Optimization (World Scientific, River Edge, NJ).Google Scholar
- [44] (1999) Boosting algorithms as gradient descent in function space. Proc. 12th Internat. Conf. Neural Inform. Processing Systems, NIPS’99 (MIT Press, Cambridge, MA), 512–518.Google Scholar
- [45] (2006) Variational Analysis and Generalized Differentiation I: Basic Theory, Grundlehren der mathematischen Wissenschaften (Springer, Berlin Heidelberg).Crossref, Google Scholar
- [46] (2012) An inexact Gauss-Newton method for inversion of basal sliding and rheology parameters in a nonlinear Stokes ice sheet model. J. Glaciology 58(211):889–903.Crossref, Google Scholar
- [47] (2003) Probabilistic programming. Ruszczynski A, Shapiro A, eds. Stochastic programming, Handbooks in Operations Research and Management Science, vol. 10 (Elsevier, Amsterdam), 267–35.Google Scholar
- [48] (2013) The fundamental risk quadrangle in risk management, optimization and statistical estimation. Surveys Oper. Res. Management Sci. 18(12):33–53.Google Scholar
- [49] (1998) Variational Analysis, A Series of Comprehensive Studies in Mathematics, vol. 317 (Springer-Verlag, Berlin Heidelberg).Google Scholar
- [50] (2006) Optimization of convex risk functions. Math. Oper. Res. 31(3):433–452.Link, Google Scholar
- [51] (2007) Corrigendum to: “Optimization of convex risk functions” [Math. Oper. Res. (2006) 31(3):, 433–452]. Math. Oper. Res. 32(2):496.Google Scholar
- [52] (1992) A version of the bundle idea for minimizing a nonsmooth function: Conceptual idea, convergence analysis, numerical results. SIAM J. Optim. 2(1):121–152.Google Scholar
- [53] (2009) On the nature and treatment of uncertainties in aerodynamic design. AIAA J. 47(3):646–654.Crossref, Google Scholar
- [54] (2004) On a class of minimax stochastic programs. SIAM J. Optim. 14(4):1237–1249.Crossref, Google Scholar
- [55] (2002) Minimax analysis of stochastic problems. Optim. Methods Software 17(3):523–542.Crossref, Google Scholar
- [56] (2014) Lectures on Stochastic Programming: Modeling and Theory, 2nd ed. MOS-SIAM Series on Optimization (SIAM, Philadelphia).Google Scholar
- [57] (1985) Minimization Methods for Non-differentiable Functions (Springer-Verlag, New York).Crossref, Google Scholar
- [58] (2010) The dynamics of plate tectonics and mantle flow: From local to global scales. Science 329(5995):1033–1038.Google Scholar
- [59] (1996) The operation of infimal convolution (Instytut Matematyczny Polskiej Akademi Nauk, Warszawa, Poland).Google Scholar
- [60] (2003) Stochastic averaging of nonlinear flows in heterogeneous porous media. J. Fluid Mech. 492:47–62.Crossref, Google Scholar
- [61] (2017) Stochastic quasi-Newton methods for nonconvex stochastic optimization. SIAM J. Optim. 27(2):927–956.Crossref, Google Scholar

