Towards the Biconjugate of Bivariate Piecewise Quadratic Functions

Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)

Abstract

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.

Keywords

Conjugate Convex envelope Piecewise quadratic function

References

1. 1.
Al-Khayyal, F.A., Falk, J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8(2), 273–286 (1983)
2. 2.
Anstreicher, K.M.: On convex relaxations for quadratically constrained quadratic programming. Math. Program. 136(2), 233–251 (2012)
3. 3.
Bauschke, H.H., Goebel, R., Lucet, Y., Wang, X.: The proximal average: basic theory. SIAM J. Optim. 19(2), 766–785 (2008)
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)
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)
7. 7.
Crama, Y.: Recognition problems for special classes of polynomials in 0–1 variables. Math. Program. 44(1–3), 139–155 (1989)
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)
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)
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)
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)
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)
15. 15.
Linderoth, J.: A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs. Math. Program. 103(2), 251–282 (2005)
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)
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)
18. 18.
Lucet, Y.: A fast computational algorithm for the Legendre-Fenchel transform. Comput. Optim. Appl. 6(1), 27–57 (1996)
19. 19.
Lucet, Y.: Faster than the fast Legendre transform, the linear-time Legendre transform. Numer. Algorithms 16(2), 171–185 (1997)
20. 20.
Lucet, Y.: Fast Moreau envelope computation i: Numerical algorithms. Numer. Algorithms 43(3), 235–249 (2006)
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)
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)
23. 23.
McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part iconvex underestimating problems. Math. Program. 10(1), 147–175 (1976)
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)
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)

© Springer Nature Switzerland AG 2020

Authors and Affiliations

• Deepak Kumar
• 1
• Yves Lucet
• 1
1. 1.University of British Columbia OkanaganKelownaCanada