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Coderivatives of Maximal Monotone Operators

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Variational Analysis and Applications

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Abstract

In this chapter we employ the tools of variational analysis and generalized differentiation developed above to study global and local monotonicity of set-valued operators.

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References

  1. F. J. Aragón Artacho and M. H. Geoffroy (2008), Characterizations of metric regularity of subdifferentials, J. Convex Anal. 15, 365–380.

    Google Scholar 

  2. F. J. Aragón Artacho and M. H. Geoffroy (2014), Metric subregularity of the convex subdifferential in Banach spaces, J. Nonlinear Convex Anal. 15, 35–47.

    Google Scholar 

  3. F. J. Aragón Artacho and B. S. Mordukhovich (2011), Enhanced metric regularity and Lipschitzian properties of variational systems, J. Global Optim. 50, 145–167.

    Google Scholar 

  4. H. H. Bauschke and P. L. Combettes (2017), Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edition, Springer, New York.

    Book  Google Scholar 

  5. J. F. Bonnans (1994), Local analysis of Newton-type methods for variational inequalities and nonlinear programming, Appl. Math. Optim. 29, 161–186.

    Google Scholar 

  6. J. M. Borwein and J. D. Vanderwerff (2010), Convex Functions, Cambridge University Press, Cambridge, United Kingdom.

    MATH  Google Scholar 

  7. R. S. Burachik and A. N. Iusem (2008), Set-Valued Mappings and Enlargements of Monotone Operators, Springer, New York.

    MATH  Google Scholar 

  8. N. H. Chieu and L. V. Hien (2017), Computation of graphical derivative for a class of normal cone mappings under a very weak condition, SIAM J. Optim. 27, 190–204.

    Google Scholar 

  9. N. H. Chieu and N. Q. Huy (2011), Second-order subdifferentials and convexity of real-valued functions, Nonlinear Anal. 74, 154–160.

    Google Scholar 

  10. N. H. Chieu, G. M. Lee, B. S. Mordukhovich and T. T. A. Nghia (2016), Coderivative characterizations of maximal monotonicity for set-valued mappings, J. Convex Anal. 23, 461–480.

    Google Scholar 

  11. N. H. Chieu and N. T. Q. Trang (2012), Coderivative and monotonicity of continuous mappings, Taiwanese J. Math. 16, 353–365.

    Google Scholar 

  12. R. Cibulka, A. D. Dontchev and A. Y. Kruger (2018), Strong metric subregularity of mappings in variational analysis and optimization, J. Math. Anal. Appl. 457, 1247–1282.

    Google Scholar 

  13. C. Ding, D. Sun and L. Zhang (2017), Characterization of the robust isolated calmness for a class of conic programming problems, SIAM J. Optim. 27, 67–90.

    Google Scholar 

  14. A. L. Dontchev (1995), Characterizations of Lipschitz stability in optimization, in Recent Developments in Well-Posed Variational Problems, edited by R. Lucchetti and J. Revalski, pp. 95–115, Kluwer, Dordrecht, The Netherlands.

    Chapter  Google Scholar 

  15. A. L. Dontchev and R. T. Rockafellar (1996), Characterizations of strong regularity for variational inequalities over polyhedral convex sets, SIAM J. Optim. 6, 1087–1105.

    Google Scholar 

  16. A. L. Dontchev and R. T. Rockafellar (1997), Characterizations of Lipschitzian stability in nonlinear programming, in Mathematical Programming with Data Perturbations, edited by A. V. Fiacco, pp. 65–82, Marcel Dekker, New York.

    MATH  Google Scholar 

  17. A. L. Dontchev and R. T. Rockafellar (2014), Implicit Functions and Solution Mappings: A View from Variational Analysis, 2nd edition, Springer, New York.

    MATH  Google Scholar 

  18. D. Drusvyatskiy and A. D. Ioffe (2015), Quadratic growth and critical point stability of semi-algebraic functions, Math. Program. 153, 635–653.

    Google Scholar 

  19. D. Drusvyatskiy and A. S. Lewis (2013), Tilt stability, uniform quadratic growth, and strong metric regularity of the subdifferential, SIAM J. Optim. 23, 256–267.

    Google Scholar 

  20. D. Drusvyatskiy, B. S. Mordukhovich and T. T. A. Nghia (2014), Second-order growth, tilt stability, and metric regularity of the subdifferential, J. Convex Anal. 21, 1165–1192.

    Google Scholar 

  21. H. Gfrerer and J. V. Outrata (2016), On computation of generalized derivatives of the normal-cone mapping and their applications, Math. Oper. Res. 41, 1535–1556.

    Google Scholar 

  22. A. F. Izmailov and M. V. Solodov (2012), Stabilized SQP revisited, Math. Program. 133, 93–120.

    Google Scholar 

  23. P. Kenderov (1975), Semi-continuity of set-valued monotone mappings, Fundamenta Mathematicae 88, 61–69.

    Google Scholar 

  24. A. J. King and R. T. Rockafellar (2002), Sensitivity analysis for nonsmooth generalized equations, Math. Program. 55, 193–212.

    Google Scholar 

  25. D. Klatte and B. Kummer (2002), Nonsmooth Equations in Optimization: Regularity, Calculus, Methods, and Applications, Kluwer, Boston, Massachusetts.

    MATH  Google Scholar 

  26. A. Y. Kruger (2009), About stationarity and regularity in variational analysis, Taiwanese J. Math. 13, 1737–1785.

    Google Scholar 

  27. B. Kummer (1991), Lipschitzian inverse functions, directional derivatives and applications in C 1,1 optimization, J. Optim. Theory Appl. 70, 561–582.

    Google Scholar 

  28. A. B. Levy (1996), Implicit multifunction theorems for the sensitivity analysis of variational conditions, Math. Program. 74, 333–350.

    Google Scholar 

  29. A. B. Levy and R. A. Poliquin (1997), Characterizing the single-valuedness of multifunctions, Set-Valued Anal. 5, 351–364.

    Google Scholar 

  30. A. B. Levy, R. A. Poliquin and R. T. Rockafellar (2000), Stability of locally optimal solutions, SIAM J. Optim. 10, 580–604.

    Google Scholar 

  31. B. S. Mordukhovich (2006), Variational Analysis and Generalized Differentiation, I: Basic Theory, Springer, Berlin.

    Google Scholar 

  32. B. S. Mordukhovich (2018), Second-Order Variational Analysis and Applications, book in progress.

    Google Scholar 

  33. B. S. Mordukhovich and T. T. A. Nghia (2013), Second-order variational analysis and characterizations of tilt-stable optimal solutions in infinite-dimensional spaces, Nonlinear Anal. 86, 159–180.

    Google Scholar 

  34. B. S. Mordukhovich and T. T. A. Nghia (2016), Local monotonicity and full stability of parametric variational systems, SIAM J. Optim. 26, 1032–1059.

    Google Scholar 

  35. B. S. Mordukhovich, J. V. Outrata and H. Ramírez C. (2015), Second-order variational analysis in conic programming with applications to optimality conditions and stability, SIAM J. Optim. 25, 76–101.

    Google Scholar 

  36. B. S. Mordukhovich, J. V. Outrata and H. Ramírez C. (2015), Graphical derivatives and stability analysis for parameterized equilibria with conic constraints, Set-Valued Var. Anal. 23, 687–704.

    Google Scholar 

  37. B. S. Mordukhovich and M. E. Sarabi (2017), Critical multipliers in variational systems via second-order generalized differentiation, Math. Program., DOI 10.1007/s10107-017-1155-2.

    Article  MATH  Google Scholar 

  38. J. V. Outrata and H. Ramírez C. (2011), On the Aubin property of critical points to perturbed second-order cone programs, SIAM J. Optim. 21, 798–823; Erratum (with F. Opazo), SIAM J. Optim. 27 (2017), 2143–2151.

    Google Scholar 

  39. J.-S. Pang (1993), Convergence of splitting and Newton methods for complementarity problems: an application of some sensitivity results, Math. Program. 58, 149–160.

    Google Scholar 

  40. T. Pennanen (2002), Local convergence of the proximal point algorithm and multiplier methods without monotonicity, Math. Oper. Res. 27, 170–191.

    Google Scholar 

  41. R. A. Poliquin and R. T. Rockafellar (1996), Prox-regular functions in variational analysis, Trans. Amer. Math. Soc. 348, 1805–1838.

    Google Scholar 

  42. R. A. Poliquin and R. T. Rockafellar (1998), Tilt stability of a local minimum, SIAM J. Optim. 8, 287–299.

    Google Scholar 

  43. S. M. Robinson (1980), Strongly regular generalized equations, Math. Oper. Res. 5, 43–62.

    Google Scholar 

  44. R. T. Rockafellar (1970), On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149, 75–88.

    Google Scholar 

  45. R. T. Rockafellar (1982), Favorable classes of Lipschitz continuous functions in subgradient optimization, in Progress in Nondifferentiable Optimization, edited by E. A. Nurminskii, pp. 125–143, IIASA, Laxenburg, Austria.

    Google Scholar 

  46. R. T. Rockafellar (1989), Proto-differentiability of set-valued mappings and its applications in optimization, Ann. Inst. H. Poincaré: Analyse Non Linéaire 6, 448–482.

    Google Scholar 

  47. R. T. Rockafellar and R. J-B. Wets (1998), Variational Analysis, Springer, Berlin.

    Book  Google Scholar 

  48. L. Thibault (1983), Tangent, cones and quasi-interiorly tangent cones to multifunctions, Trans. Amer. Math. Soc. 277, 601–621.

    Google Scholar 

  49. A. Uderzo (2016), A strong metric regularity analysis of nonsmooth mappings via steepest displacement rate, J. Optim. Theory Appl. 171, 573–599.

    Google Scholar 

  50. E. H. Zarantonello (1960), Solving functional equations by contractive averaging, University of Wisconsin, MRC Report 160, Madison, Wisconsin.

    Google Scholar 

  51. E. H. Zarantonello (1971), Projections on convex sets in Hilbert space and spectral theory, I: projections on convex sets, Contributions to Nonlinear Functional Analysis, pp. 237–341, Academic Press, New York.

    Google Scholar 

  52. R. Zhang and J. S. Treiman (1995), Upper-Lipschitz multifunctions and inverse subdifferentials, Nonlinear Anal. 24, 273–286.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Wayne State University, Detroit, MI, USA

    Boris S. Mordukhovich

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  1. Boris S. Mordukhovich
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Mordukhovich, B.S. (2018). Coderivatives of Maximal Monotone Operators. In: Variational Analysis and Applications. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-92775-6_5

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