Towards the Biconjugate of Bivariate Piecewise Quadratic Functions

  • Deepak Kumar
  • Yves LucetEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)


Computing the closed convex envelope or biconjugate is the core operation that bridges the domain of nonconvex with convex analysis. We focus here on computing the conjugate of a bivariate piecewise quadratic function defined over a polytope. First, we compute the convex envelope of each piece, which is characterized by a polyhedral subdivision such that over each member of the subdivision, it has a rational form (square of a linear function over a linear function). Then we compute the conjugate of all such rational functions. It is observed that the conjugate has a parabolic subdivision such that over each member of its subdivision, it has a fractional form (linear function over square root of a linear function). This computation of the conjugate is performed with a worst-case linear time complexity algorithm. Our results are an important step toward computing the conjugate of a piecewise quadratic function, and further in obtaining explicit formulas for the convex envelope of piecewise rational functions.


Conjugate Convex envelope Piecewise quadratic function 


  1. 1.
    Al-Khayyal, F.A., Falk, J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8(2), 273–286 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Anstreicher, K.M.: On convex relaxations for quadratically constrained quadratic programming. Math. Program. 136(2), 233–251 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bauschke, H.H., Goebel, R., Lucet, Y., Wang, X.: The proximal average: basic theory. SIAM J. Optim. 19(2), 766–785 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bauschke, H.H., Lucet, Y., Trienis, M.: How to transform one convex function continuously into another. SIAM Rev. 50(1), 115–132 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brenier, Y.: Un algorithme rapide pour le calcul de transformées de Legendre-Fenchel discretes. Comptes rendus de l’Académie des sciences. Série 1, Mathématique 308(20), 587–589 (1989)Google Scholar
  6. 6.
    Corrias, L.: Fast Legendre-Fenchel transform and applications to Hamilton-Jacobi equations and conservation laws. SIAM J. Numer. Anal. 33(4), 1534–1558 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Crama, Y.: Recognition problems for special classes of polynomials in 0–1 variables. Math. Program. 44(1–3), 139–155 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Gardiner, B., Lucet, Y.: Graph-matrix calculus for computational convex analysis. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 243–259. Springer (2011)Google Scholar
  11. 11.
    Gardiner, B., Lucet, Y.: Computing the conjugate of convex piecewise linear-quadratic bivariate functions. Math. Program. 139(1–2), 161–184 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Haque, T., Lucet, Y.: A linear-time algorithm to compute the conjugate of convex piecewise linear-quadratic bivariate functions. Comput. Optim. Appl. 70(2), 593–613 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hiriart-Urruty, J.B., Lemaréchal, C.: Convex analysis and minimization algorithms II: Advanced Theory and Bundle Methods. Springer Science & Business Media (1993)Google Scholar
  14. 14.
    Jach, M., Michaels, D., Weismantel, R.: The convex envelope of (n-1)-convex functions. SIAM J. Optim. 19(3), 1451–1466 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Linderoth, J.: A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs. Math. Program. 103(2), 251–282 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Locatelli, M.: A technique to derive the analytical form of convex envelopes for some bivariate functions. J. Glob. Optim. 59(2–3), 477–501 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Locatelli, M.: Polyhedral subdivisions and functional forms for the convex envelopes of bilinear, fractional and other bivariate functions over general polytopes. J. Glob. Optim. 66(4), 629–668 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Lucet, Y.: A fast computational algorithm for the Legendre-Fenchel transform. Comput. Optim. Appl. 6(1), 27–57 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Lucet, Y.: Faster than the fast Legendre transform, the linear-time Legendre transform. Numer. Algorithms 16(2), 171–185 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Lucet, Y.: Fast Moreau envelope computation i: Numerical algorithms. Numer. Algorithms 43(3), 235–249 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Lucet, Y.: What shape is your conjugate? A survey of computational convex analysis and its applications. SIAM Rev. 52(3), 505–542 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Lucet, Y., Bauschke, H.H., Trienis, M.: The piecewise linear-quadratic model for computational convex analysis. Comput. Optim. Appl. 43(1), 95–118 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part iconvex underestimating problems. Math. Program. 10(1), 147–175 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, vol. 317. Springer Science & Business Media (1998)Google Scholar
  25. 25.
    Sherali, H.D., Alameddine, A.: An explicit characterization of the convex envelope of a bivariate bilinear function over special polytopes. Ann. Oper. Res. 25(1), 197–209 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Tardella, F.: On the existence of polyhedral convex envelopes. In: Frontiers in Global Optimization, pp. 563–573. Springer (2004)Google Scholar
  27. 27.
    Tardella, F.: Existence and sum decomposition of vertex polyhedral convex envelopes. Optim. Lett. 2(3), 363–375 (2008)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.University of British Columbia OkanaganKelownaCanada

Personalised recommendations