Epi-convergence: The Moreau Envelope and Generalized Linear-Quadratic Functions



This work explores the class of generalized linear-quadratic functions, constructed using maximally monotone symmetric linear relations. Calculus rules and properties of the Moreau envelope for this class of functions are developed. In finite dimensions, on a metric space defined by Moreau envelopes, we consider the epigraphical limit of a sequence of quadratic functions and categorize the results. We examine the question of when a quadratic function is a Moreau envelope of a generalized linear-quadratic function; characterizations involving nonexpansiveness and Lipschitz continuity are established. This work generalizes some results by Hiriart-Urruty and by Rockafellar and Wets.


Attouch–Wets metric Complete metric space Epi-convergence Extended seminorm Fenchel conjugate Firmly nonexpansive Generalized linear-quadratic function Linear relation Lipschitz continuous Maximally monotone Nonexpansive Moreau envelope Proximal mapping 

Mathematics Subject Classification

47A06 52A41 47H05 90C31 



The authors would like to thank the editor and the anonymous referee for their very helpful comments and suggestions for improvement of this manuscript. Chayne Planiden was supported by UBC University Graduate Fellowship and by Natural Sciences and Engineering Research Council of Canada. Xianfu Wang was partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant.


  1. 1.
    Moreau, J.J.: Proximité et dualité dans un espace Hilbertien. Bull. Soc. Math. Fr. 93, 273–299 (1965)CrossRefMATHGoogle Scholar
  2. 2.
    Poliquin, R., Rockafellar, R.: Generalized Hessian properties of regularized nonsmooth functions. SIAM J. Optim. 6(4), 1121–1137 (1996)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Attouch, H., Wets, R.: Quantitative stability of variational systems: I. The epigraphical distance. Trans. Am. Math. Soc. 328, 695–729 (1991)MathSciNetMATHGoogle Scholar
  4. 4.
    Attouch, H.: Variational Convergence for Functions and Operators. Applicable Mathematics Series. Pitman, Boston (1984)MATHGoogle Scholar
  5. 5.
    Cross, R.: Multivalued Linear Operators. Monographs and Textbooks in Pure and Applied Mathematics, vol. 213. Marcel Dekker Inc, New York (1998)MATHGoogle Scholar
  6. 6.
    Bartz, S., Bauschke, H., Moffat, S., Wang, X.: The resolvent average of monotone operators: dominant and recessive properties. SIAM J. Optim. 26(1), 602–634 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bauschke, H., Wang, X., Yao, L.: Monotone linear relations: maximality and Fitzpatrick functions. J. Convex Anal. 16(3–4), 673–686 (2009)MathSciNetMATHGoogle Scholar
  8. 8.
    Bauschke, H., Wang, X., Yao, L.: On Borwein–Wiersma decompositions of monotone linear relations. SIAM J. Optim. 20(5), 2636–2652 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Yao, L.: On Monotone Linear Relations and the Sum Problem in Banach Spaces. UBC Ph. D. thesis (2011)Google Scholar
  10. 10.
    Lemaréchal, C., Sagastizábal, C.: Practical aspects of the Moreau–Yosida regularization: theoretical preliminaries. SIAM J. Optim. 7(2), 367–385 (1997)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Rockafellar, R.: Convex Analysis. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1997)MATHGoogle Scholar
  12. 12.
    Hiriart-Urruty, J.B.: The deconvolution operation in convex analysis: an introduction. Cybern. Syst. Anal. 30(4), 555–560 (1994)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Rockafellar, R., Wets, R.: Variational Analysis. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  14. 14.
    Bauschke, H., Combettes, P.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)CrossRefMATHGoogle Scholar
  15. 15.
    Borwein, J., Vanderwerff, J.: Convex Functions: Constructions, Characterizations and Counterexamples. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2010)CrossRefMATHGoogle Scholar
  16. 16.
    Attouch, H., Beer, G.: On the convergence of subdifferentials of convex functions. Arch. Math. (Basel) 60(4), 389–400 (1993)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Burke, J., Hoheisel, T.: Epi-convergent smoothing with applications to convex composite functions. SIAM J. Optim. 23(3), 1457–1479 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Rockafellar, R., Royset, J.: Random variables, monotone relations, and convex analysis. Math. Program. 148(1–2, Ser. B), 297–331 (2014)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Wijsman, R.A.: Convergence of sequences of convex sets, cones and functions. II. Trans. Am. Math. Soc. 123, 32–45 (1966)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Beer, G.: Topologies on Closed and Closed Convex Sets. Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht (1993)CrossRefMATHGoogle Scholar
  21. 21.
    Planiden, C., Wang, X.: Strongly convex functions, Moreau envelopes and the generic nature of convex functions with strong minimzers. SIAM J. Optim. 26(2), 1341–1364 (2016)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Bartz, S., Bauschke, H., Borwein, J., Reich, S., Wang, X.: Fitzpatrick functions, cyclic monotonicity and Rockafellar’s antiderivative. Nonlinear Anal. 66(5), 1198–1223 (2007)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Baillon, J.B., Haddad, G.: Quelques propriétés des opérateurs angle-bornés et \(n\)-cycliquement monotones. Isr. J. Math. 26(2), 137–150 (1977)CrossRefMATHGoogle Scholar
  24. 24.
    Bauschke, H., Borwein, J., Wang, X., Yao, L.: The Brezis–Browder theorem in a general Banach space. J. Funct. Anal. 262(12), 4948–4971 (2012)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms II: Advanced Theory and Bundle Methods, vol. 306. Springer, Berlin (2013)MATHGoogle Scholar
  26. 26.
    Meyer, C.: Matrix Analysis and Applied Linear Algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2000)CrossRefGoogle Scholar
  27. 27.
    Mordukhovich, B.: Variational Analysis and Generalized Differentiation I: Basic Theory, vol. 330. Springer, Berlin (2006)Google Scholar
  28. 28.
    Roman, S.: Advanced Linear Algebra, vol. 3. Springer, Berlin (2005)MATHGoogle Scholar
  29. 29.
    Beer, G.: Norms with infinite values. J. Convex Anal. 22(1), 35–58 (2015)MathSciNetMATHGoogle Scholar
  30. 30.
    Beer, G., Vanderwerff, J.: Structural properties of extended normed spaces. Set-Valued Var. Anal. 23(4), 613–630 (2015)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics & Applied StatisticsUniversity of WollongongWollongongAustralia
  2. 2.Department of MathematicsUniversity of British Columbia OkanaganKelownaCanada

Personalised recommendations