Inner Moreau Envelope of Nonsmooth Conic Chance-Constrained Optimization Problems

References

• [1] Aliprantis CD, Border KC (2006) Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd ed. (Springer, Berlin).Google Scholar
• [2] Alvarez O, Barron EN, Ishii H (1999) Hopf-Lax formulas for semicontinuous data. Indiana Univ. Math. J. 48(3):993–1035.Crossref, Google Scholar
• [3] Attouch H (1984) Variational Convergence for Functions and Operators, Applicable Mathematics Series (Pitman Publishing, Boston).Google Scholar
• [4] Attouch H, Azé D (1993) Approximation and regularization of arbitrary functions in Hilbert spaces by the Lasry-Lions method. Ann. Inst. Henri Poincaré Anal. Non Linéaire 10(3):289–312.Crossref, Google Scholar
• [5] Attouch H, Buttazzo G, Michaille G (2014) Variational Analysis in Sobolev and BV Spaces, MOS-SIAM Series on Optimization, vol. 17 (Society for Industrial and Applied Mathematics, Philadelphia).Google Scholar
• [6] Balder EJ, Sambucini AR (2005) Fatou’s lemma for multifunctions with unbounded values in a dual space. J. Convex Anal. 12(2):383–395.Google Scholar
• [7] Bauschke HH, Combettes PL (2017) Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics (Springer, Cham, Switzerland).Google Scholar
• [8] Beck A (2017) First-Order Methods in Optimization, MOS-SIAM Series on Optimization, vol. 25 (Society for Industrial and Applied Mathematics, Philadelphia).Google Scholar
• [9] Bonnans JF, Shapiro A (2000) Perturbation Analysis of Optimization Problems, Springer Series in Operations Research (Springer-Verlag, New York).Google Scholar
• [10] Borwein JM, Vanderwerff JD (2010) Convex Functions: Constructions, Characterizations and Counterexamples, Encyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, UK).Google Scholar
• [11] Boyd S, El Ghaoui L, Feron E, Balakrishnan V (1994) Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, vol 15 (Society for Industrial and Applied Mathematics, Philadelphia).Google Scholar
• [12] Bremer I, Henrion R, Möller A (2015) Probabilistic constraints via SQP solver: Application to a renewable energy management problem. Comput. Management Sci. 12:435–459.Crossref, Google Scholar
• [13] Correa R, Hantoute A, Pérez-Aros P (2019) Characterizations of the subdifferential of convex integral functions under qualification conditions. J. Functional Anal. 277(1):227–254.Crossref, Google Scholar
• [14] Correa R, Hantoute A, Pérez-Aros P (2021) Qualification conditions—Free characterizations of the ε-subdifferential of convex integral functions. Appl. Math. Optim. 83(3):1709–1737.Crossref, Google Scholar
• [15] Correa R, López MA, Pérez-Aros P (2021) Necessary and sufficient optimality conditions in DC semi-infinite programming. SIAM J. Optim. 31(1):837–865.Crossref, Google Scholar
• [16] de Oliveira W (2020) The ABC of DC programming. Set-Valued Variational Anal. 28:679–706.Crossref, Google Scholar
• [17] de Oliveira W, Sagastizábal C (2014) Level bundle methods for oracles with on demand accuracy. Optim. Methods Software 29(6):1180–1209.Crossref, Google Scholar
• [18] Elstrodt J (2011) Maß und Integrationstheorie, 7th ed. (Springer-Verlag).Google Scholar
• [19] Fabian M, Habala P, Hájek P, Montesinos V, Zizler V (2011) Banach Space Theory: The Basis for Linear and Nonlinear Analysis, CMS Books in Mathematics (Springer, New York).Google Scholar
• [20] Fang K, Kotz S, Ng KW (1990) Symmetric Multivariate and Related Distributions, Monographs on Statistics and Applied Probability, vol. 36 (Springer-Science).Google Scholar
• [21] Farshbaf-Shaker MH, Henrion R, Hömberg D (2018) Properties of chance constraints in infinite dimensions with an application to PDE constrained optimization. Set-Valued Variational Anal. 26(4):821–841.Crossref, Google Scholar
• [22] Geletu A, Hoffmann A, Klöppel M, Li P (2017) An inner-outer approximation approach to chance constrained optimization. SIAM J. Optim. 27(3):1834–1857.Crossref, Google Scholar
• [23] Guo S, Xu H, Zhang L (2017) Convergence analysis for mathematical programs with distributionally robust chance constraint. SIAM J. Optim. 27(2):784–816.Crossref, Google Scholar
• [24] Hantoute A, Henrion R, Pérez-Aros P (2019) Subdifferential characterization of probability functions under Gaussian distribution. Math. Programming 174(1–2):167–194.Crossref, Google Scholar
• [25] Heitsch H (2020) On probabilistic capacity maximization in a stationary gas network. Optim. 69(3):575–604.Crossref, Google Scholar
• [26] Henrion R, Strugarek C (2008) Convexity of chance constraints with independent random variables. Comput. Optim. Appl. 41:263–276.Crossref, Google Scholar
• [27] Hiriart-Urruty JB (1985) Generalized differentiability, duality and optimization for problems dealing with differences of convex functions. Convexity and Duality in Optimization (Groningen, 1984), Lecture Notes in Economics and Mathematical Systems, vol. 256 (Springer, Berlin), 37–70.Crossref, Google Scholar
• [28] Hong L, Yang Y, Zhang L (2011) Sequential convex approximations to joint chance constrained programed: A Monte Carlo approach. Oper. Res. 3(59):617–630.Link, Google Scholar
• [29] Ishii H (1999) Hopf-Lax formulas for Hamilton-Jacobi equations with semicontinuous initial data. Sūrikaisekikenkyūsho Kōkyūroku (1111):144–156.Google Scholar
• [30] Laguel Y, van Ackooij W, Malick J, Matiussi Ramalho G (2022) On the convexity of level-sets of probability functions. J. Convex Anal. 29(2):1–32.Google Scholar
• [31] Marti K (1995) Differentiation of probability functions: The transformation method. Comput. Math. Appl. 30:361–382.Crossref, Google Scholar
• [32] Mayer J (2000) On the numerical solution of jointly chance constrained problems. Uryas’ev S, ed. Probabilistic Constrained Optimization: Methodology and Applications (Kluwer Academic Publishers), 220–235.Google Scholar
• [33] Mordukhovich BS (2006) Variational Analysis and Generalized Differentiation I Basic Theory, Grundlehren der Mathematischen Wissenschaften, vol. 330 (Springer-Verlag, Berlin).Google Scholar
• [34] Mordukhovich BS (2018) Variational Analysis and Applications, Springer Monographs in Mathematics (Springer, Cham, Switzerland).Google Scholar
• [35] Nemirovski A, Shapiro A (2006) Convex approximations of chance constrained programs. SIAM J. Optim. 17(4):969–996.Crossref, Google Scholar
• [36] Nesterov Y, Nemirovskii A (1994) Interior-Point Polynomial Algorithms in Convex Programming, SIAM Studies in Applied Mathematics, vol. 13 (Society for Industrial and Applied Mathematics, Philadelphia).Google Scholar
• [37] Oliveira W, Sagastizábal C, Lemaréchal C (2014) Convex proximal bundle methods in depth: A unified analysis for inexact oracles. Math. Programmming Ser. B 148:241–277.Crossref, Google Scholar
• [38] Peña-Ordieres A, Luedtke JR, Wächter A (2020) Solving chance-constrained problems via a smooth sample-based nonlinear approximation. SIAM J. Optim. 30(3):2221–2250.Crossref, Google Scholar
• [39] Pérez-Aros P (2019) Formulae for the conjugate and the subdifferential of the supremum function. J. Optim. Theory Appl. 180(2):397–427.Crossref, Google Scholar
• [40] Pérez-Aros P (2019) Subdifferential formulae for the supremum of an arbitrary family of functions. SIAM J. Optim. 29(2):1714–1743.Crossref, Google Scholar
• [41] Pérez-Aros P, Vilches E (2021) Moreau envelope of supremum functions with applications to infinite and stochastic programming. SIAM J. Optim. 31(3):1635–1657.Crossref, Google Scholar
• [42] Pólik I, Terlaky T (2007) A survey of the S-lemma. SIAM Rev. 49(3):371–418.Crossref, Google Scholar
• [43] Poljak S, Rendl F, Wolkowicz H (1995) A recipe for semidefinite relaxation for (0, 1)-quadratic programming. J. Global Optim. 7(1):51–73.Crossref, Google Scholar
• [44] Pommelet A (1982) Analyse convexe et théorie de Morse. Unpublished PhD thesis, Université de Paris IX, France.Google Scholar
• [45] Prékopa A (1971) Logarithmic concave measures with applications to stochastic programming. Acta Scientiarium Mathematicarum (Szeged) 32:301–316.Google Scholar
• [46] Rockafellar RT, Wets RJB (1998) Variational Analysis, Grundlehren der Mathematischen Wissenschaften, vol. 317 (Springer-Verlag, Berlin).Google Scholar
• [47] Rosen JB (1965) Pattern separation by convex programming. J. Math. Anal. Appl. 10:123–134.Crossref, Google Scholar
• [48] Royset J, Polak E (2004) Implementable algorithm for stochastic optimization using sample average approximations. J. Optim. Theory Appl. 122(1):157–184.Crossref, Google Scholar
• [49] Shan F, Xiao XT, Zhang LW (2016) Convergence analysis on a smoothing approach to joint chance constrained programs. Optim. 65(12):2171–2193.Crossref, Google Scholar
• [50] Shan F, Zhang L, Xiao X (2014) A smoothing function approach to joint chance constrained programs. J. Optim. Theory Appl. 163:181–199.Crossref, Google Scholar
• [51] Shapiro A, Dentcheva D, Ruszczyński A (2014) Lectures on Stochastic Programming, MOS-SIAM Series on Optimization, vol. 9 (Society for Industrial and Applied Mathematics, Philadelphia).Google Scholar
• [52] Szántai T (1988) A computer code for solution of probabilistic-constrained stochastic programming problems. Ermoliev Y, Wets RJ-B, eds. Numerical Techniques for Stochastic Optimization, 229–235.Crossref, Google Scholar
• [53] Tuy H (2016) Convex Analysis and Global Optimization, Nonconvex Optimization and Its Applications, vol. 22 (Springer).Google Scholar
• [54] Uryas’ev S (1995) Derivatives of probability functions and some applications. Ann. Oper. Res. 56:287–311.Crossref, Google Scholar
• [55] van Ackooij W (2015) Eventual convexity of chance constrained feasible sets. Optim. 64(5):1263–1284.Crossref, Google Scholar
• [56] van Ackooij W (2020) A discussion of probability functions and constraints from a variational perspective. Set-Valued Variational Anal. 28(4):585–609.Crossref, Google Scholar
• [57] van Ackooij W, de Oliveira W (2016) Convexity and optimization with copulæstructured probabilistic constraints. Optim. 65(7):1349–1376.Crossref, Google Scholar
• [58] van Ackooij W, Henrion R (2014) Gradient formulae for nonlinear probabilistic constraints with Gaussian and Gaussian-like distributions. SIAM J. Optim. 24(4):1864–1889.Crossref, Google Scholar
• [59] van Ackooij W, Henrion R (2017) (Sub-)gradient formulae for probability functions of random inequality systems under Gaussian distribution. SIAM J. Uncertainty Quant. 5(1):63–87.Crossref, Google Scholar
• [60] van Ackooij W, Malick J (2016) Decomposition algorithm for large-scale two-stage unit-commitment. Ann. Oper. Res. 238(1):587–613.Crossref, Google Scholar
• [61] van Ackooij W, Malick J (2019) Eventual convexity of probability constraints with elliptical distributions. Math. Programming 175(1):1–27.Crossref, Google Scholar
• [62] van Ackooij W, Pérez-Aros P (2019) Generalized differentiation of probability functions acting on an infinite system of constraints. SIAM J. Optim. 29(3):2179–2210.Crossref, Google Scholar
• [63] van Ackooij W, Pérez-Aros P (2020) Gradient formulae for nonlinear probabilistic constraints with non-convex quadratic forms. J. Optim. Theory Appl. 185(1):239–269.Crossref, Google Scholar
• [64] van Ackooij W, Pérez-Aros P (2021) Gradient formulae for probability functions depending on a heterogenous family of constraints. Open J. Math. Optim. 2:1–29.Crossref, Google Scholar
• [65] van Ackooij W, Pérez-Aros P (2022) Generalized differentiation of probability functions: Parameter dependent sets given by intersections of convex sets and complements of convex sets. Appl. Math. Optim. 85(2):1–39.Google Scholar
• [66] van Ackooij W, Sagastizábal C (2014) Constrained bundle methods for upper inexact oracles with application to joint chance constrained energy problems. SIAM J. Optim. 24(2):733–765.Crossref, Google Scholar
• [67] van Ackooij W, Aleksovska I, Zuniga MM (2018) (Sub-)differentiability of probability functions with elliptical distributions. Set-Valued Variational Anal. 26(4):887–910.Crossref, Google Scholar
• [68] van Ackooij W, Henrion R, Pérez-Aros P (2020) Generalized gradients for probabilistic/robust (probust) constraints. Optim. 69(7–8):1451–1479.Crossref, Google Scholar
• [69] van Ackooij W, Javal P, Pérez-Aros P (2022) Derivatives of probability functions: Unions of polyhedra and elliptical distributions. Set-Valued Variational Anal. 30(2):487–519.Crossref, Google Scholar
• [70] Vandenberghe L, Boyd S (1996) Semidefinite programming. SIAM Rev. 38(1):49–95.Crossref, Google Scholar