Journal of Optimization Theory and Applications

, Volume 148, Issue 1, pp 107–124 | Cite as

Convexity of the Proximal Average

  • Jennifer A. Johnstone
  • Valentin R. Koch
  • Yves Lucet
Article

Abstract

We complete the study of the convexity of the proximal average by proving it is convex as a function of each of its parameters separately, but not jointly convex as a function of any two of its parameters. We present an interpolation-based plotting algorithm that takes advantage of the partial convexity of the proximal average, and improves the plotting time by a factor of 100, while reducing picture sizes by a factor of 10.

Keywords

Convex analysis Proximal average Convexity Interpolation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Atteia, M., Raïssouli, M.: Self dual operators on convex functionals; geometric mean and square root of convex functionals. J. Convex Anal. 8, 223–240 (2001) MATHMathSciNetGoogle Scholar
  2. 2.
    Bauschke, H.H., Wang, X.: The kernel average of two convex functions and its application to the extension and representation of monotone operators. Trans. Am. Math. Soc. 361, 5947–5965 (2009) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bauschke, H.H., Matoušková, E., Reich, S.: Projection and proximal point methods: Convergence results and counterexamples. Nonlinear Anal. 56, 715–738 (2004) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bauschke, H.H., Lucet, Y., Wang, X.: Primal-dual symmetric intrinsic methods for finding antiderivatives of cyclically monotone operators. SIAM J. Control Optim. 46, 2031–2051 (2007) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bauschke, H.H., Goebel, R., Lucet, Y., Wang, X.: The proximal average: Basic theory. SIAM J. Optim. 19, 768–785 (2008) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Bauschke, H.H., Lucet, Y., Trienis, M.: How to transform one convex function continuously into another. SIAM Rev. 50, 115–132 (2008) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Ceria, S., Soares, J.: Convex programming for disjunctive convex optimization. Math. Program. 86, 595–614 (1999) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gardiner, B., Lucet, Y.: Numerical computation of Fitzpatrick functions. J. Convex Anal. 16, 779–790 (2009) MATHMathSciNetGoogle Scholar
  9. 9.
    Ghoussoub, N.: A theory of anti-selfdual Lagrangians: stationary case. C. R. Math. Acad. Sci. Paris 340, 245–250 (2005) MATHMathSciNetGoogle Scholar
  10. 10.
    Ghoussoub, N.: Maximal monotone operators are selfdual vector fields and vice-versa (2006). http://www.birs.ca/-02-08.pdf
  11. 11.
    Ghoussoub, N.: Selfdual Partial Differential Systems and their Variational Principles. Springer, Berlin (2009) Google Scholar
  12. 12.
    Goebel, R.: Self-dual smoothing of convex and saddle functions. J. Convex Anal. 15, 179–190 (2008) MATHMathSciNetGoogle Scholar
  13. 13.
    Hare, W.: A proximal average for nonconvex functions: A proximal stability perspective. SIAM J. Optim. 20, 650–666 (2009) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 305–306. Springer, Berlin (1993). Vol. I: Fundamentals, Vol. II: Advanced theory and bundle methods Google Scholar
  15. 15.
    Lucet, Y.: A fast computational algorithm for the Legendre–Fenchel transform. Comput. Optim. Appl. 6, 27–57 (1996) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lucet, Y.: Faster than the fast Legendre transform, the linear-time Legendre transform. Numer. Algorithms 16, 171–185 (1997) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Lucet, Y.: Fast Moreau envelope computation I: Numerical algorithms. Numer. Algorithms 43, 235–249 (2006) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lucet, Y.: New sequential exact Euclidean distance transform algorithms based on convex analysis. Image Vis. Comput. 27, 37–44 (2009) CrossRefGoogle Scholar
  19. 19.
    Lucet, Y.: What shape is your conjugate? A survey of computational convex analysis and its applications. SIAM J. Optim. 20, 216–250 (2009) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Lucet, Y., Bauschke, H.H., Trienis, M.: The piecewise linear-quadratic model for computational convex analysis. Comput. Optim. Appl. 43, 95–118 (2009) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Maréchal, P.: On a functional operation generating convex functions. I. Duality. J. Optim. Theory Appl. 126, 175–189 (2005) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Maréchal, P.: On a functional operation generating convex functions. II. Algebraic properties. J. Optim. Theory Appl. 126, 357–366 (2005) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    McAllister, D.F., Roulier, J.A.: Interpolation by convex quadratic splines. Math. Comput. 32, 1154–1162 (1978) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    McAllister, D.F., Roulier, J.A.: An algorithm for computing a shape-preserving osculatory quadratic spline. ACM Trans. Math. Softw. 7, 331–347 (1981) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Moreau, J.J.: Proximité et dualité dans un espace Hilbertien. Bull. Soc. Math. France 93, 273–299 (1965) MATHMathSciNetGoogle Scholar
  26. 26.
    Renka, R.J.: Algorithm 790: CSHEP2D: cubic Shepard method for bivariate interpolation of scattered data. ACM Trans. Math. Softw. 25, 70–73 (1999) MATHCrossRefGoogle Scholar
  27. 27.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998) MATHCrossRefGoogle Scholar
  28. 28.
    Segre, E.: Enrico color graphic toolbox. http://www.scilab.org/contrib/ (2005)

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Jennifer A. Johnstone
    • 1
  • Valentin R. Koch
    • 1
  • Yves Lucet
    • 2
  1. 1.Mathematics, Irving K. Barber SchoolUniversity of British Columbia OkanaganKelownaCanada
  2. 2.Computer Science, Irving K. Barber SchoolUniversity of British Columbia OkanaganKelownaCanada

Personalised recommendations