Computing the partial conjugate of convex piecewise linear-quadratic bivariate functions

  • Bryan Gardiner
  • Khan Jakee
  • Yves Lucet


Piecewise linear-quadratic (PLQ) functions are an important class of functions in convex analysis since the result of most convex operators applied to a PLQ function is a PLQ function. We modify a recent algorithm for computing the convex (Legendre-Fenchel) conjugate of convex PLQ functions of two variables, to compute its partial conjugate i.e. the conjugate with respect to one of the variables. The structure of the original algorithm is preserved including its time complexity (linear time with some approximation and log-linear time without approximation). Applying twice the partial conjugate (and a variable switching operator) recovers the full conjugate. We present our partial conjugate algorithm, which is more flexible and simpler than the original full conjugate algorithm. We emphasize the difference with the full conjugate algorithm and illustrate results by computing partial conjugates, partial Moreau envelopes, and partial proximal averages.


Legendre-Fenchel transform Convex conjugate Piecewise linear-quadratic functions Computational convex analysis Partial conjugate 



Yves Lucet and Khan Jakee were partially supported by a Discovery grant from the Natural Sciences and Engineering Research Council of Canada (NSERC). Bryan Gardiner was partially supported by an NSERC Undergraduate Student Research Award. Part of the research was performed in the Computer-Aided Convex Analysis laboratory funded by the Canadian Foundation for Innovation.

Special thanks go to Rafal Goebel who pointed out the proof of Fact 3.3 in [20]. The authors are indebted to the referees for invaluable feedback that resulted in the inclusion of Fact 3.3, and greatly improved the presentation of the paper.


  1. 1.
    Alfeld, P., Schumaker, L.L.: Smooth macro-elements based on Powell-Sabin triangle splits. Adv. Comput. Math. 16, 29–46 (2002) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bauschke, H.H., Goebel, R., Lucet, Y., Wang, X.: The proximal average: basic theory. SIAM J. Optim. 19, 768–785 (2008) CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bauschke, H.H., Lucet, Y., Trienis, M.: How to transform one convex function continuously into another. SIAM Rev. 50, 115–132 (2008) CrossRefMATHMathSciNetGoogle 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) CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bauschke, H.H., Moffat, S.M., Wang, X.: Self-dual smooth approximations of convex functions via the proximal average. In: Bauschke, H.H., Burachik, R.S., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Optimization and Its Applications, vol. 49, pp. 23–32. Springer, New York (2011) CrossRefGoogle Scholar
  6. 6.
    Bauschke, H.H., Wang, X., Yao, L.: Autoconjugate representers for linear monotone operators. Math. Program. 123, 5–24 (2010) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Brenier, Y.: Un algorithme rapide pour le calcul de transformées de Legendre–Fenchel discrètes. C. R. Acad. Sci. Paris, Sér. I, Math. 308, 587–589 (1989) MATHMathSciNetGoogle Scholar
  8. 8.
    Cao, J., Li, X., Wang, G., Qin, H.: Surface reconstruction using bivariate simplex splines on Delaunay configurations. In: IEEE International Conference on Shape Modeling and Applications. Computers & Graphics-UK, Tsinghua Univ, Beijing, P.R. China, Jun. 26–28, 2009, vol. 33, pp. 341–350 (2009) Google Scholar
  9. 9.
    CGAL: Computational geometry algorithms library.
  10. 10.
    CGLAB a Scilab toolbox for geometry based on CGAL.
  11. 11.
    Consortium, S.: Scilab (1994).
  12. 12.
    Corrias, L.: Fast Legendre–Fenchel transform and applications to Hamilton–Jacobi equations and conservation laws. SIAM J. Numer. Anal. 33, 1534–1558 (1996) CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Dæhlen, M., Lyche, T.: Bivariate interpolation with quadratic box splines. Math. Comput. 51, 219–230 (1988) MATHGoogle Scholar
  14. 14.
    de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry, 3rd edn. Algorithms and Applications. Springer, Berlin (2008) MATHGoogle Scholar
  15. 15.
    Dierckx, P., Van Leemput, S., Vermeire, T.: Algorithms for surface fitting using Powell-Sabin splines. IMA J. Numer. Anal. 12, 271–299 (1992) CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Gardiner, B., Lucet, Y.: Numerical computation of Fitzpatrick functions. J. Convex Anal. 16, 779–790 (2009) MATHMathSciNetGoogle Scholar
  17. 17.
    Gardiner, B., Lucet, Y.: Convex hull algorithms for piecewise linear-quadratic functions in computational convex analysis. Set-Valued Var. Anal. 18, 467–482 (2010) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Gardiner, B., Lucet, Y.: Graph-matrix calculus for computational convex analysis. In: Bauschke, H.H., Burachik, R.S., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Optimization and Its Applications, vol. 49, pp. 243–259. Springer, New York (2011) CrossRefGoogle Scholar
  19. 19.
    Gardiner, B., Lucet, Y.: Computing the conjugate of convex piecewise linear-quadratic bivariate functions. Math. Program. 1–24 (2013) Google Scholar
  20. 20.
    Goebel, R.: Convexity, convergence and feedback in optimal control. PhD thesis, University of Washington, Seattle (2000) Google Scholar
  21. 21.
    Goebel, R.: Self-dual smoothing of convex and saddle functions. J. Convex Anal. 15, 179–190 (2008) MATHMathSciNetGoogle Scholar
  22. 22.
    Hare, W.: A proximal average for nonconvex functions: a proximal stability perspective. SIAM J. Optim. 20, 650–666 (2009) CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    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
  24. 24.
    Johnstone, J., Koch, V., Lucet, Y.: Convexity of the proximal average. J. Optim. Theory Appl. 148, 107–124 (2011) CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Lai, M.-J., Schumaker, L.L.: Spline Functions on Triangulations. Encyclopedia of Mathematics and Its Applications, vol. 110. Cambridge University Press, Cambridge (2007) CrossRefMATHGoogle Scholar
  26. 26.
    Lucet, Y.: A fast computational algorithm for the Legendre–Fenchel transform. Comput. Optim. Appl. 6, 27–57 (1996) CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Lucet, Y.: Computational convex analysis library, 1996–2011.
  28. 28.
    Lucet, Y.: Faster than the fast Legendre transform, the linear-time Legendre transform. Numer. Algorithms 16, 171–185 (1997) CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Lucet, Y.: Fast Moreau envelope computation I: numerical algorithms. Numer. Algorithms 43, 235–249 (2006) CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Lucet, Y.: What shape is your conjugate? A survey of computational convex analysis and its applications. SIAM Rev. 52, 505–542 (2010) CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Lucet, Y., Bauschke, H.H., Trienis, M.: The piecewise linear-quadratic model for computational convex analysis. Comput. Optim. Appl. 43, 95–118 (2009) CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Manni, C., Sablonnière, P.: Quadratic spline quasi-interpolants on Powell-Sabin partitions. Adv. Comput. Math. 26, 283–304 (2007) CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Moffat, S.M.: On the kernel average of n functions. Master’s thesis, Department of Mathematics, University of British Columbia (Dec 2009) Google Scholar
  34. 34.
    Noullez, A., Vergassola, M.: A fast Legendre transform algorithm and applications to the adhesion model. J. Sci. Comput. 9, 259–281 (1994) CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Patrinos, P., Sarimveis, H.: Convex parametric piecewise quadratic optimization: theory, algorithms and control applications. Tech. rep., National Technical University of Athens, Greece (2010) Google Scholar
  36. 36.
    Patrinos, P., Sarimveis, H.: A new algorithm for solving convex parametric quadratic programs based on graphical derivatives of solution mappings. Automatica 46, 1405–1418 (2010) CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Patrinos, P., Sarimveis, H.: Convex parametric piecewise quadratic optimization: theory and algorithms. Automatica 47, 1770–1777 (2011) CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Penot, J.-P.: The relevance of convex analysis for the study of monotonicity. Nonlinear Anal. 58, 855–871 (2004) CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Powell, M.J.D., Sabin, M.A.: Piecewise quadratic approximations on triangles. ACM Trans. Math. Softw. 3, 316–325 (1977) CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970) MATHGoogle Scholar
  41. 41.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998) CrossRefMATHGoogle Scholar
  42. 42.
    Sbibih, D., Serghini, A., Tijini, A.: Polar forms and quadratic spline quasi-interpolants on Powell-Sabin partitions. Appl. Numer. Math. 59, 938–958 (2009) CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Sbibih, D., Serghini, A., Tijini, A.: Bivariate simplex spline quasi-interpolants. Numer. Math. 3, 97–118 (2010) MATHMathSciNetGoogle Scholar
  44. 44.
    She, Z.-S., Aurell, E., Frisch, U.: The inviscid Burgers equation with initial data of Brownian type. Commun. Math. Phys. 148, 623–641 (1992) CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    Sorokina, T., Zeilfelder, F.: Optimal quasi-interpolation by quadratic C 1-splines on type-2 triangulations. In: Approximation Theory XI: Gatlinburg 2004, Mod. Methods Math., pp. 423–438. Nashboro Press, Brentwood (2005) Google Scholar
  46. 46.
    Speleers, H., Dierckx, P., Vandewalle, S.: Quasi-hierarchical Powell-Sabin B-splines. Comput. Aided Geom. Des. 26, 174–191 (2009) CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    Wiley, D.F., Childs, H.R., Hamann, B., Joy, K.I., Max, N.L.: Best quadratic spline approximation for hierarchical visualization. In: Proceedings of the Symposium on Data Visualisation 2002, VISSYM ’02, Aire-la-Ville. Eurographics Association, Switzerland, pp. 133–140 (2002) Google Scholar
  48. 48.
    Willemans, K., Dierckx, P.: Surface fitting using convex Powell-Sabin splines. J. Comput. Appl. Math. 56, 263–282 (1994) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Computer Science, I. K. Barber SchoolUniversity of British Columbia OkanaganKelownaCanada

Personalised recommendations