Building Formulations for Piecewise Linear Relaxations of Nonlinear Functions

Published Online:https://doi.org/10.1287/opre.2023.0187

References

  • Ahmadi H, Martí JR, Moshref A (2013) Piecewise linear approximation of generators cost functions using max-affine functions. 2013 IEEE Power & Energy Soc. General Meeting (IEEE, Piscataway, NJ), 1–5.CrossrefGoogle Scholar
  • Balas E (1975) Disjunctive programming: Cutting planes from logical conditions. Mangasarian OL, Meyer RR, Robinson SM, eds. Nonlinear Programming 2 (Academic Press, Cambridge, MA), 279–312.CrossrefGoogle Scholar
  • Balas E (1979) Disjunctive programming. Hammer PL, Johnson EL, Korte BH, eds. Discrete Optimization II, Annals of Discrete Mathematics, vol. 5 (Elsevier, Amsterdam), 3–51.CrossrefGoogle Scholar
  • Balas E (1998) Disjunctive programming: Properties of the convex hull of feasible points. Discrete Appl. Math. 89(1–3):3–44.CrossrefGoogle Scholar
  • Beale EML, Tomlin JA (1970) Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables. Oper. Res. 69(447–454):99.Google Scholar
  • Benders JF (1962) Partitioning procedures for solving mixed-variables programming problems. Numer. Math. 4(1):238–252.CrossrefGoogle Scholar
  • Bergamini ML, Aguirre P, Grossmann I (2005) Logic-based outer approximation for globally optimal synthesis of process networks. Comput. Chemical Engrg. 29(9):1914–1933.CrossrefGoogle Scholar
  • Bergamini ML, Grossmann I, Scenna N, Aguirre P (2008) An improved piecewise outer-approximation algorithm for the global optimization of MINLP models involving concave and bilinear terms. Comput. Chemical Engrg. 32(3):477–493.CrossrefGoogle Scholar
  • Bertsimas D, Mišić VV (2017) Robust product line design. Oper. Res. 65(1):19–37.LinkGoogle Scholar
  • Bestuzheva K, Besançon M, Chen WK, Chmiela A, Donkiewicz T, van Doornmalen J, Eifler L, et al. (2021) The SCIP optimization suite 8.0. Preprint, submitted December 16, https://arxiv.org/abs/2112.08872.Google Scholar
  • Bezanson J, Edelman A, Karpinski S, Shah VB (2017) Julia: A fresh approach to numerical computing. SIAM Rev. 59(1):65–98.CrossrefGoogle Scholar
  • Bondy JA, Murty USR (2008) Graph Theory (Springer, London).CrossrefGoogle Scholar
  • Braun K, Burlacu R (2023) A computational study for piecewise linear relaxations of mixed-integer nonlinear programs. Accessed December 3, 2024, https://optimization-online.org/2023/09/a-computational-study-for-piecewise-linear-relaxations-of-mixed-integer-nonlinear-programs.Google Scholar
  • Camm JD, Cochran JJ, Curry DJ, Kannan S (2006) Conjoint optimization: An exact branch-and-bound algorithm for the share-of-choice problem. Management Sci. 52(3):435–447.LinkGoogle Scholar
  • Camponogara E, Nazari LF (2015) Models and algorithms for optimal piecewise-linear function approximation. Math. Problems Engrg. 2015(1):876862.CrossrefGoogle Scholar
  • Castillo Castillo PA, Castro PM, Mahalec V (2018) Global optimization of MIQCPs with dynamic piecewise relaxations. J. Global Optim. 71:691–716.CrossrefGoogle Scholar
  • Castro PM (2015) Tightening piecewise McCormick relaxations for bilinear problems. Comput. Chemical Engrg. 72:300–311.CrossrefGoogle Scholar
  • Castro PM (2016) Normalized multiparametric disaggregation: An efficient relaxation for mixed-integer bilinear problems. J. Global Optim. 64(4):765–784.CrossrefGoogle Scholar
  • Codas A, Camponogara E (2012) Mixed-integer linear optimization for optimal lift-gas allocation with well-separator routing. Eur. J. Oper. Res. 217(1):222–231.CrossrefGoogle Scholar
  • Codas A, Campos S, Camponogara E, Gunnerud V, Sunjerga S (2012) Integrated production optimization of oil fields with pressure and routing constraints: The Urucu field. Comput. Chemical Engrg. 46:178–189.CrossrefGoogle Scholar
  • Codsi J, Ngueveu SU, Gendron B (2025) LinA: A faster approach to piecewise linear approximations using corridors and its application to mixed-integer optimization. Math. Programming Comput. https://doi.org/10.1007/s12532-024-00274-8.Google Scholar
  • Croxton KL, Gendron B, Magnanti TL (2003) A comparison of mixed-integer programming models for nonconvex piecewise linear cost minimization problems. Management Sci. 49(9):1268–1273.LinkGoogle Scholar
  • Dai H, Izatt G, Tedrake R (2019) Global inverse kinematics via mixed-integer convex optimization. Internat. J. Robotics Res. 38(12–13):1420–1441.CrossrefGoogle Scholar
  • D’Ambrosio C, Lodi A, Martello S (2010) Piecewise linear approximation of functions of two variables in MILP models. Oper. Res. Lett. 38(1):39–46.CrossrefGoogle Scholar
  • Dantzig GB, Wolfe P (1960) Decomposition principle for linear programs. Oper. Res. 8(1):101–111.LinkGoogle Scholar
  • de Farias IR, Zhao M Jr, Zhao H (2008) A special ordered set approach for optimizing a discontinuous separable piecewise linear function. Oper. Res. Lett. 36(2):234–238.CrossrefGoogle Scholar
  • de Farias I, Kozyreff E, Gupta R, Zhao M (2013) Branch-and-cut for separable piecewise linear optimization and intersection with semi-continuous constraints. Math. Programming Comp. 5(1):75–112.CrossrefGoogle Scholar
  • de la Torre P, Greenlaw R, Schäffer AA (1995) Optimal edge ranking of trees in polynomial time. Algorithmica 13(6):592–618.CrossrefGoogle Scholar
  • Deits R, Tedrake R (2014) Footstep planning on uneven terrain with mixed-integer convex optimization. 2014 IEEE-RAS Internat. Conf. Humanoid Robots (IEEE, Piscataway, NJ), 279–286.Google Scholar
  • Dong H, Du Z (2015) Obstacle avoidance path planning of planar redundant manipulators using workspace density. Internat. J. Adv. Robotic Systems 12(2):9.CrossrefGoogle Scholar
  • Dunning I, Huchette J, Lubin M (2017) JuMP: A modeling language for mathematical optimization. SIAM Rev. 59(2):295–320.CrossrefGoogle Scholar
  • Geißler B, Martin A, Morsi A, Schewe L (2012) Using piecewise linear functions for solving MINLPs. Lee J, Leyffer S, eds. Mixed Integer Nonlinear Programming, The IMA Volumes in Mathematics and Its Applications, vol. 154 (Springer, New York), 287–314.CrossrefGoogle Scholar
  • Gurobi Optimization, LLC (2023) Gurobi optimizer reference manual. https://www.gurobi.com.Google Scholar
  • He T, Tawarmalani M (2021) A new framework to relax composite functions in nonlinear programs. Math. Programming 190:427–466.CrossrefGoogle Scholar
  • He T, Tawarmalani M (2022) Tractable relaxations of composite functions. Math. Oper. Res. 47(2):1110–1140.LinkGoogle Scholar
  • He T, Tawarmalani M (2024) MIP relaxations in factorable programming. SIAM J. Optim. 34(3):2856–2882.CrossrefGoogle Scholar
  • Huchette J, Vielma JP (2019) A combinatorial approach for small and strong formulations of disjunctive constraints. Math. Oper. Res. 44(3):793–820.LinkGoogle Scholar
  • Huchette J, Vielma JP (2022) Nonconvex piecewise linear functions: Advanced formulations and simple modeling tools. Oper. Res. 71(5):1835–1856.Google Scholar
  • Iyer AV, Ratliff HD, Vijayan G (1991) On an edge ranking problem of trees and graphs. Discrete Appl. Math. 30(1):43–52.CrossrefGoogle Scholar
  • Jeroslow RG, Lowe JK (1984) Modelling with integer variables. Korte B, Ritter K, eds. Mathematical Programming at Oberwolfach II, Mathematical Programming Studies, vol. 22 (Springer, Berlin, Heidelberg), 167–184.CrossrefGoogle Scholar
  • Jeroslow RG, Lowe JK (1985) Experimental results on the new techniques for integer programming formulations. J. Oper. Res. Soc. 36(5):393–403.CrossrefGoogle Scholar
  • Keha AB, de Farias IR Jr, Nemhauser GL (2004) Models for representing piecewise linear cost functions. Oper. Res. Lett. 32(1):44–48.CrossrefGoogle Scholar
  • Keha AB, de Farias IR Jr, Nemhauser GL (2006) A branch-and-cut algorithm without binary variables for nonconvex piecewise linear optimization. Oper. Res. 54(5):847–858.LinkGoogle Scholar
  • Kim J, Richard JPP, Tawarmalani M (2024) Piecewise polyhedral relaxations of multilinear optimization. SIAM J. Optim. 34(4):3167–3193.CrossrefGoogle Scholar
  • Kucuk S, Bingul Z (2006) Robot Kinematics: Forward and Inverse Kinematics (INTECH Open Access Publisher, London).Google Scholar
  • Lam TW, Yue FL (2001) Optimal edge ranking of trees in linear time. Algorithmica 30(1):12–33.CrossrefGoogle Scholar
  • Lyu B, Hicks IV (2023) Maximal clique and edge-ranking bounds of biclique cover number. Preprint, submitted February 24, https://arxiv.org/abs/2302.12775.Google Scholar
  • Lyu B, Hicks IV, Huchette J (2024) Modeling combinatorial disjunctive constraints via junction trees. Math. Programming 204:385–413.Google Scholar
  • McCormick GP (1976) Computability of global solutions to factorable nonconvex programs: Part I—Convex underestimating problems. Math. Programming 10(1):147–175.CrossrefGoogle Scholar
  • Minkowski H (1897) Allgemeine lehrsatze uber die konvexen polyeder. Nachr. Ges. Wiss. Gottingen, Math.-Phys. KL. 1897(1897):198–219.Google Scholar
  • Misener R, Floudas CA (2012) Global optimization of mixed-integer quadratically-constrained quadratic programs (MIQCQP) through piecewise-linear and edge-concave relaxations. Math. Programming 136(1):155–182.CrossrefGoogle Scholar
  • Misener R, Thompson JP, Floudas CA (2011) APOGEE: Global optimization of standard, generalized, and extended pooling problems via linear and logarithmic partitioning schemes. Comput. Chemical Engrg. 35(5):876–892.CrossrefGoogle Scholar
  • Ngueveu SU (2019) Piecewise linear bounding of univariate nonlinear functions and resulting mixed integer linear programming-based solution methods. Eur. J. Oper. Res. 275(3):1058–1071.CrossrefGoogle Scholar
  • O’Rourke J (1998) Computational Geometry in C, 2nd ed. (Cambridge University Press, Cambridge, UK).CrossrefGoogle Scholar
  • Padberg M (2000) Approximating separable nonlinear functions via mixed zero-one programs. Oper. Res. Lett. 27(1):1–5.CrossrefGoogle Scholar
  • Rebennack S, Kallrath J (2015) Continuous piecewise linear delta-approximations for univariate functions: Computing minimal breakpoint systems. J. Optim. Theory Appl. 167(2):617–643.CrossrefGoogle Scholar
  • Rebennack S, Krasko V (2020) Piecewise linear function fitting via mixed-integer linear programming. INFORMS J. Comput. 32(2):507–530.LinkGoogle Scholar
  • Silva TL, Camponogara E (2014) A computational analysis of multidimensional piecewise-linear models with applications to oil production optimization. Eur. J. Oper. Res. 232(3):630–642.CrossrefGoogle Scholar
  • Sundar K, Sanjeevi S, Nagarajan H (2022) Sequence of polyhedral relaxations for nonlinear univariate functions. Optim. Engrg. 23(2):877–894.CrossrefGoogle Scholar
  • Sundar K, Nagarajan H, Linderoth J, Wang S, Bent R (2021) Piecewise polyhedral formulations for a multilinear term. Oper. Res. Lett. 49(1):144–149.CrossrefGoogle Scholar
  • Trindade RS, d’Ambrosio C, Frangioni A, Gentile C (2023) Comparing perspective reformulations for piecewise-convex optimization. Oper. Res. Lett. 51(6):702–708.CrossrefGoogle Scholar
  • Vielma JP (2018) Embedding formulations and complexity for unions of polyhedra. Management Sci. 64(10):4721–4734.LinkGoogle Scholar
  • Vielma JP, Nemhauser GL (2011) Modeling disjunctive constraints with a logarithmic number of binary variables and constraints. Math. Programming 128(1):49–72.CrossrefGoogle Scholar
  • Vielma JP, Ahmed S, Nemhauser G (2010) Mixed-integer models for nonseparable piecewise-linear optimization: Unifying framework and extensions. Oper. Res. 58(2):303–315.LinkGoogle Scholar
  • Wang X, Camm JD, Curry DJ (2009) A branch-and-price approach to the share-of-choice product line design problem. Management Sci. 55(10):1718–1728.LinkGoogle Scholar
  • Weyl H (1934) Elementare theorie der konvexen polyeder. Commentarii Math. Helvetici 7(1):290–306.CrossrefGoogle Scholar
  • Yahya S, Moghavvemi M, Mohamed HA (2011) Geometrical approach of planar hyper-redundant manipulators: Inverse kinematics, path planning and workspace. Simulation Modeling Practice Theory 19(1):406–422.CrossrefGoogle Scholar
  • Zhou X, Nishizeki T (1995) Finding optimal edge-rankings of trees. Proc. Sixth Annual ACM-SIAM Sympos. Discrete Algorithms (Society for Industrial and Applied Mathematics, Philadelphia), 122–131.Google Scholar
INFORMS site uses cookies to store information on your computer. Some are essential to make our site work; Others help us improve the user experience. By using this site, you consent to the placement of these cookies. Please read our Privacy Statement to learn more.