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.
Supported by NSERC, CFI.
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References
Al-Khayyal, F.A., Falk, J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8(2), 273–286 (1983)
Anstreicher, K.M.: On convex relaxations for quadratically constrained quadratic programming. Math. Program. 136(2), 233–251 (2012)
Bauschke, H.H., Goebel, R., Lucet, Y., Wang, X.: The proximal average: basic theory. SIAM J. Optim. 19(2), 766–785 (2008)
Bauschke, H.H., Lucet, Y., Trienis, M.: How to transform one convex function continuously into another. SIAM Rev. 50(1), 115–132 (2008)
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)
Corrias, L.: Fast Legendre-Fenchel transform and applications to Hamilton-Jacobi equations and conservation laws. SIAM J. Numer. Anal. 33(4), 1534–1558 (1996)
Crama, Y.: Recognition problems for special classes of polynomials in 0–1 variables. Math. Program. 44(1–3), 139–155 (1989)
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)
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)
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)
Gardiner, B., Lucet, Y.: Computing the conjugate of convex piecewise linear-quadratic bivariate functions. Math. Program. 139(1–2), 161–184 (2013)
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)
Hiriart-Urruty, J.B., Lemaréchal, C.: Convex analysis and minimization algorithms II: Advanced Theory and Bundle Methods. Springer Science & Business Media (1993)
Jach, M., Michaels, D., Weismantel, R.: The convex envelope of (n-1)-convex functions. SIAM J. Optim. 19(3), 1451–1466 (2008)
Linderoth, J.: A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs. Math. Program. 103(2), 251–282 (2005)
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)
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)
Lucet, Y.: A fast computational algorithm for the Legendre-Fenchel transform. Comput. Optim. Appl. 6(1), 27–57 (1996)
Lucet, Y.: Faster than the fast Legendre transform, the linear-time Legendre transform. Numer. Algorithms 16(2), 171–185 (1997)
Lucet, Y.: Fast Moreau envelope computation i: Numerical algorithms. Numer. Algorithms 43(3), 235–249 (2006)
Lucet, Y.: What shape is your conjugate? A survey of computational convex analysis and its applications. SIAM Rev. 52(3), 505–542 (2010)
Lucet, Y., Bauschke, H.H., Trienis, M.: The piecewise linear-quadratic model for computational convex analysis. Comput. Optim. Appl. 43(1), 95–118 (2009)
McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part iconvex underestimating problems. Math. Program. 10(1), 147–175 (1976)
Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, vol. 317. Springer Science & Business Media (1998)
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)
Tardella, F.: On the existence of polyhedral convex envelopes. In: Frontiers in Global Optimization, pp. 563–573. Springer (2004)
Tardella, F.: Existence and sum decomposition of vertex polyhedral convex envelopes. Optim. Lett. 2(3), 363–375 (2008)
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Kumar, D., Lucet, Y. (2020). Towards the Biconjugate of Bivariate Piecewise Quadratic Functions. In: Le Thi, H., Le, H., Pham Dinh, T. (eds) Optimization of Complex Systems: Theory, Models, Algorithms and Applications. WCGO 2019. Advances in Intelligent Systems and Computing, vol 991. Springer, Cham. https://doi.org/10.1007/978-3-030-21803-4_27
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DOI: https://doi.org/10.1007/978-3-030-21803-4_27
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