Skip to main content
SpringerLink
Log in

Visualization of the \(\varepsilon \)-subdifferential of piecewise linear–quadratic functions

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

Computing explicitly the \(\varepsilon \)-subdifferential of a proper function amounts to computing the level set of a convex function namely the conjugate minus a linear function. The resulting theoretical algorithm is applied to the the class of (convex univariate) piecewise linear–quadratic functions for which existing numerical libraries allow practical computations. We visualize the results in a primal, dual, and subdifferential views through several numerical examples. We also provide a visualization of the Brøndsted–Rockafellar theorem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Aravkin, A., Burke, J., Pillonetto, G.: Sparse/robust estimation and Kalman smoothing with nonsmooth log-concave densities: modeling, computation, and theory. J. Mach. Learn. Res. 14, 2689–2728 (2013). http://jmlr.org/papers/v14/aravkin13a.html

  2. Bonnans, J., Gilbert, J.C., Lemaréchal, C., Sagastizábal, C.A.: Numerical Optimization: Theoretical and Practical Aspects. Springer, Berlin (2006)

    MATH  Google Scholar 

  3. Borwein, J., Lewis, A.: Convex Analysis and Nonlinear Optimization: Theory and Examples. Springer, Berlin (2010)

    Google Scholar 

  4. Brøndsted, A., Rockafellar, R.T.: On the subdifferentiability of convex functions. Proc. Am. Math. Soc. 16, 605–611 (1965). http://www.jstor.org/stable/2033889

  5. Correa, R., Hantoute, A., Jourani, A.: Characterizations of convex approximate subdifferential calculus in Banach spaces. Trans. Am. Math. Soc. 368(7), 4831–4854 (2016). doi:10.1090/tran/6589

    Article  MathSciNet  MATH  Google Scholar 

  6. Correa, R., Lemaréchal, C.: Convergence of some algorithms for convex minimization. Math. Program. 62(2 Ser. B), 261–275 (1993). doi:10.1007/BF01585170

    Article  MathSciNet  MATH  Google Scholar 

  7. Dembo, R., Anderson, R.: An efficient linesearch for convex piecewise-linear/quadratic functions. In: Advances in numerical partial differential equations and optimization (Mérida, 1989), pp. 1–8. SIAM, Philadelphia, PA (1991)

  8. de Oliveira, W., Sagastizábal, C.: Level bundle methods for oracles with on-demand accuracy. Optim. Methods Softw. 29(6), 1180–1209 (2014). doi:10.1080/10556788.2013.871282

    Article  MathSciNet  MATH  Google Scholar 

  9. de Oliveira, W., Solodov, M.: A doubly stabilized bundle method for nonsmooth convex optimization. Math. Program. 156(1–2, Ser. A), 125–159 (2016). doi:10.1007/s10107-015-0873-6

    Article  MathSciNet  MATH  Google Scholar 

  10. Frangioni, A.: Generalized bundle methods. SIAM J. Optim. 13(1), 117–156 (2002). doi:10.1137/S1052623498342186. (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gardiner, B., Jakee, K., Lucet, Y.: Computing the partial conjugate of convex piecewise linear-quadratic bivariate functions. Comput. Optim. Appl. 58(1), 249–272 (2014). doi:10.1007/s10589-013-9622-z

    Article  MathSciNet  MATH  Google Scholar 

  12. Gardiner, B., Lucet, Y.: Convex hull algorithms for piecewise linear-quadratic functions in computational convex analysis. Set Valued Var. Anal. 18(3–4), 467–482 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gardiner, B., Lucet, Y.: Computing the conjugate of convex piecewise linear-quadratic bivariate functions. Math. Program. 139(1–2), 161–184 (2013). doi:10.1007/s10107-013-0666-8

    Article  MathSciNet  MATH  Google Scholar 

  14. Hare, W., Planiden, C.: Thresholds of prox-boundedness of PLQ functions. J. Convex Anal. 23(3), 1–28 (2016)

    MathSciNet  MATH  Google Scholar 

  15. Hare, W., Sagastizábal, C.: A redistributed proximal bundle method for nonconvex optimization. SIAM J. Optim. 20(5), 2442–2473 (2010). doi:10.1137/090754595

    Article  MathSciNet  MATH  Google Scholar 

  16. Hiriart-Urruty, J., Lemaréchal, C.: Convex Analysis and Minimization Algorithms II: Advanced Theory and Bundle Methods, vol. 306 of Grundlehren der mathematischen Wissenschaften. Springer, New York (1993)

    MATH  Google Scholar 

  17. Hiriart-Urruty, J., Moussaoui, M., Seeger, A., Volle, M.: Subdifferential calculus without qualification conditions, using approximate subdifferentials: A survey. Nonlinear Anal. Theory Methods Appl. 24(12), 1727–1754 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ioffe, A.D.: Approximate subdifferentials and applications. I. The finite-dimensional theory. Trans. Am. Math. Soc. 281(1), 389–416 (1984). doi:10.2307/1999541

    MathSciNet  MATH  Google Scholar 

  19. Jakee, K.M.K.: Computational Convex Analysis Using Parametric Quadratic Programming. Master’s thesis, University of British Columbia (2013). https://circle.ubc.ca/handle/2429/45182

  20. Kiwiel, K.: Proximity control in bundle methods for convex nondifferentiable minimization. Math. Program. 46(1, (Ser. A)), 105–122 (1990). doi:10.1007/BF01585731

    Article  MathSciNet  MATH  Google Scholar 

  21. Kiwiel, K.: Proximal level bundle methods for convex nondifferentiable optimization, saddle-point problems and variational inequalities. Math. Program. 69(1, Ser. B), 89–109 (1995). doi:10.1007/BF01585554. Nondifferentiable and large-scale optimization (Geneva, 1992)

    MathSciNet  MATH  Google Scholar 

  22. Lemaréchal, C., Sagastizábal, C.: Variable metric bundle methods: from conceptual to implementable forms. Math. Program. 76(3, Ser. B), 393–410 (1997). doi:10.1016/S0025-5610(96)00053-6

    Article  MathSciNet  MATH  Google Scholar 

  23. Lucet, Y.: Computational convex analysis library, 1996–2016. http://atoms.scilab.org/toolboxes/CCA/

  24. Lucet, Y., Bauschke, H., Trienis, M.: The piecewise linear–quadratic model for computational convex analysis. Comput. Optim. Appl. 43(1), 95–118 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rantzer, A., Johansson, M.: Piecewise linear quadratic optimal control. IEEE Trans. Automat. Control 45(4), 629–637 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rockafellar, R.: On the Essential Boundedness of Solutions to Problems in Piecewise Linear Quadratic Optimal Control. Analyse mathematique et applications. Gauthier villars, pp. 437–443 (1988)

  27. Rockafellar, R.: Convex Analysis. Princeton University Press, Princeton (2015)

    MATH  Google Scholar 

  28. Rockafellar, R., Wets, R.: A lagrangian finite generation technique for solving linear-quadratic problems in stochastic programming. In: Stochastic Programming 84 Part II, pp. 63–93. Springer, Berlin (1986)

  29. Rockafellar, R., Wets, R.: Variational Analysis, vol. 317. Springer, Berlin (2009)

    MATH  Google Scholar 

  30. Scilab:: Scilab. http://www.scilab.org/ (2015)

  31. Trienis, M.: Computational Convex Analysis: From Continuous Deformation to Finite Convex Integration. Master thesis (2007)

Download references

Acknowledgements

This work was supported in part by Discovery Grants #355571-2013 (Hare) and #298145-2013 (Lucet) from NSERC, and The University of British Columbia, Okanagan campus. Part of the research was performed in the Computer-Aided Convex Analysis (CA2) laboratory funded by a Leaders Opportunity Fund (LOF) from the Canada Foundation for Innovation (CFI) and by a British Columbia Knowledge Development Fund (BCKDF). Special thanks to Heinz Bauschke for recommending we visualize the Brøndsted–Rockafellar theorem.

Author information

Authors and Affiliations

  1. Mathematics, 1132 Faculty/Administration Building, Wayne State University, 42 W. Warren Ave., Detroit, MI, 48202, USA

    Anuj Bajaj

  2. Mathematics, ASC 353, The University of British Columbia - Okanagan (UBCO), 3187 University Way, Kelowna, BC, V1V 1V7, Canada

    Warren Hare

  3. Computer Science, ASC 350, The University of British Columbia - Okanagan (UBCO), 3187 University Way, Kelowna, BC, V1V 1V7, Canada

    Yves Lucet

Authors
  1. Anuj Bajaj
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Warren Hare
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yves Lucet
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yves Lucet.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bajaj, A., Hare, W. & Lucet, Y. Visualization of the \(\varepsilon \)-subdifferential of piecewise linear–quadratic functions. Comput Optim Appl 67, 421–442 (2017). https://doi.org/10.1007/s10589-017-9892-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-017-9892-y

Keywords

Search

Navigation