Conditions for the existence, identification and calculus rules of the threshold of prox-boundedness


This note advances knowledge of the threshold of prox-boundedness of a function; an important concern in the use of proximal point optimization algorithms and in determining the existence of the Moreau envelope of the function. In finite dimensions, we study general prox-bounded functions and then focus on some useful classes such as piecewise functions and Lipschitz continuous functions. The thresholds are explicitly determined when possible and bounds are established otherwise. Some calculus rules are constructed; we consider functions with known thresholds and find the thresholds of their sum and composition.

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  1. 1.

    Annergren, M., Hansson, A., Wahlberg, B.: An ADMM algorithm for solving \(l_1\)-regularized MPC. In: 2012 IEEE 51st Conference on Decision and Control, pp. 4486–4491. IEEE (2012)

  2. 2.

    Attouch, H., Peypouquet, J.: Convergence of inertial dynamics and proximal algorithms governed by maximally monotone operators. Math. Program. 174(1–2, Ser. B), 391–432 (2019)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bačak, M., Borwein, J., Eberhard, A., Mordukhovich, B.: Infimal convolutions and Lipschitzian properties of subdifferentials for prox-regular functions in Hilbert spaces. J. Convex Anal. 17(3–4), 737–763 (2010)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Bauschke, H., Combettes, P.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2011)

    Google Scholar 

  5. 5.

    Bauschke, H., Matoušková, E., Reich, S.: Projection and proximal point methods: convergence results and counterexamples. Nonlinear Anal. 56(5), 715–738 (2004)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trend. Mach. Learn. 3(1), 1–122 (2011)

    Article  Google Scholar 

  7. 7.

    Briceno-Arias, L., Combettes, P., Pesquet, J.-C., Pustelnik, N.: Proximal algorithms for multicomponent image recovery problems. J. Math. Imag. Vision 41(1–2), 3–22 (2011)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Burachik, R., Iusem, A.: A generalized proximal point algorithm for the nonlinear complementarity problem. RAIRO Oper. Res. 33(4), 447–479 (1999)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Calamai, P., Moré, J.: Projected gradient methods for linearly constrained problems. Math. Program. 39(1), 93–116 (1987)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Chen, Y., Kan, C., Song, W.: The Moreau envelope function and proximal mapping with respect to the Bregman distances in Banach spaces. Vietnam J. Math. 40(2–3), 181–199 (2012)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Dao, M., Tam, M.: Union averaged operators with applications to proximal algorithms for min-convex functions. J. Optim. Theory Appl. 181(1), 61–94 (2019)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Douglas, J., Rachford, H.: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82(2), 421–439 (1956)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Eckstein, J., Bertsekas, D.: On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55(3, Ser. A), 293–318 (1992)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Fougner, C., Boyd, S.: Parameter selection and preconditioning for a graph form solver. In: Emerging Applications of Control and Systems Theory, pp. 41–61. Springer, Berlin (2018)

  15. 15.

    Fuduli, A., Gaudioso, M.: Tuning strategy for the proximity parameter in convex minimization. J. Optim. Theory Appl. 130(1), 95–112 (2006)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Fukushima, M., Mine, H.: A generalized proximal point algorithm for certain nonconvex minimization problems. Int. J. Syst. Sci. 12(8), 989–1000 (1981)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29(2), 403–419 (1991)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Hare, W.: A proximal average for nonconvex functions: a proximal stability perspective. SIAM J. Optim. 20(2), 650–666 (2009)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Hare, W., Planiden, C.: Thresholds of prox-boundedness of PLQ functions. J. Convex Anal. 23(3), 691–718 (2016)

  20. 20.

    Hare, W., Planiden, C.: Computing proximal points of convex functions with inexact subgradients. Set-Valued Var. Anal. 26(3), 469–492 (2018)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Hare, W., Planiden, C., Sagastizábal, C.: A derivative-free VU-algorithm for convex finite-max problems. arXiv preprint arXiv:1903.11184 (2019)

  22. 22.

    Hare, W., Sagastizábal, C.: Computing proximal points of nonconvex functions. Math. Program. 116(1–2, Ser. B), 221–258 (2009)

    MathSciNet  Article  Google Scholar 

  23. 23.

    He, B., Yang, H., Wang, S.: Alternating direction method with self-adaptive penalty parameters for monotone variational inequalities. J. Optim. Theory Appl. 106(2), 337–356 (2000)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Hiriart-Urruty, J.-B., Le, H.: From Eckart and Young approximation to Moreau envelopes and vice versa. RAIRO Oper. Res. 47(3), 299–310 (2013)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Kaplan, A., Tichatschke, R.: Proximal point methods and nonconvex optimization. J. Glob. Optim. 13(4), 389–406 (1998). Workshop on Global Optimization (Trier, 1997)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Kecis, I., Thibault, L.: Moreau envelopes of s-lower regular functions. Nonlinear Anal. 127, 157–181 (2015)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Li, Y., Zhang, H., Li, Z., Gao, H.: Proximal gradient method with automatic selection of the parameter by automatic differentiation. Optim. Methods Softw. 33(4–6), 708–717 (2018)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Lucet, Y.: Fast Moreau envelope computation. I. Numerical algorithms. Numer. Algorithms 43(3), 235–249 (2007)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Martinet, B.: Régularisation d’inéquations variationnelles par approximations successives. Rev. Française Informat. Recherche Opérationnelle 4(Ser. R–3), 154–158 (1970)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Mifflin, R., Sagastizábal, C.: VU-smoothness and proximal point results for some nonconvex functions. Optim. Methods Softw. 19(5), 463–478 (2004)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Mohammadi, A., Mordukhovich, B., Sarabi, M.: Variational analysis of composite models with applications to continuous optimization. arXiv preprint arXiv:1905.08837 (2019)

  32. 32.

    Moreau, J.-J.: Proximité et dualité dans un espace Hilbertien. Bull. Soc. Math. Fr. 93, 273–299 (1965)

    Article  Google Scholar 

  33. 33.

    Planiden, C., Wang, X.: Strongly convex functions, Moreau envelopes, and the generic nature of convex functions with strong minimizers. SIAM J. Optim. 26(2), 1341–1364 (2016)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Planiden, C., Wang, X.: Epi-convergence: the Moreau envelope and generalized linear-quadratic functions. J. Optim. Theory Appl. 177(1), 21–63 (2018)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Planiden, C., Wang, X.: Proximal mappings and Moreau envelopes of single-variable convex piecewise cubic functions and multivariable gauge functions. In: Nonsmooth Optimization and Its Applications, volume 170 of International Series on Numerical Mathematics, pp. 89–130. Birkhäuser/Springer, Cham (2019)

  36. 36.

    Rey, P., Sagastizábal, C.: Dynamical adjustment of the prox-parameter in bundle methods. Optimization 51(2), 423–447 (2002)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Rockafellar, R.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877–898 (1976)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Rockafellar, R., Wets, R.: Variational Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1998)

    Google Scholar 

  39. 39.

    Sun, W., Sampaio, R., Candido, M.: Proximal point algorithm for minimization of DC function. J. Comput. Math. 21(4), 451–462 (2003)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Yosida, K.: Functional Analysis. Die Grundlehren der Mathematischen Wissenschaften, vol. 123. Springer/Academic Press, Berlin (1965)

    Google Scholar 

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The author wishes to thank the anonymous referees for their valued input in the improvement of this work and to Referee 1 in particular for suggesting a cleaner, more concise proof of Proposition 3.2.

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Planiden, C. Conditions for the existence, identification and calculus rules of the threshold of prox-boundedness. Optim Lett 15, 45–57 (2021).

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  • Fenchel conjugate
  • Infimal convolution
  • Lipschitz continuous
  • Moreau envelope
  • Moreau–Yosida regularization
  • Piecewise function
  • Prox-bounded
  • Proximal mapping
  • Regularization
  • Threshold