Spatial Branch-and-Bound for Nonconvex Separable Piecewise Linear Optimization

Published Online:https://doi.org/10.1287/ijoc.2024.0755

References

  • Achterberg T, Koch T, Martin A (2005) Branching rules revisited. Oper. Res. Lett. 33(1):42–54.CrossrefGoogle Scholar
  • Adams W, Gupte A, Xu Y (2019) Error bounds for monomial convexification in polynomial optimization. Math. Programming 175(1–2):355–393.CrossrefGoogle Scholar
  • Adjiman CS, Dallwig S, Floudas CA, Neumaier A (1998) A global optimization method, αBB, for general twice-differentiable constrained NLPs—I. Theoretical advances. Comput. Chemical Engrg. 22(9):1137–1158.CrossrefGoogle Scholar
  • Al-Khayyal FA, Sherali HD (2000) On finitely terminating branch-and-bound algorithms for some global optimization problems. SIAM J. Optim. 10(4):1049–1057.CrossrefGoogle Scholar
  • Barber CB, Dobkin DP, Huhdanpaa H (1996) The quickhull algorithm for convex hulls. ACM Trans. Math. Software 22(4):469–483.CrossrefGoogle Scholar
  • Bärmann A, Burlacu R, Hager L, Kleinert T (2023) On piecewise linear approximations of bilinear terms: Structural comparison of univariate and bivariate mixed-integer programming formulations. J. Global Optim. 85(4):789–819.CrossrefGoogle Scholar
  • Beach B, Hildebrand R, Huchette J (2022) Compact mixed-integer programming formulations in quadratic optimization. J. Global Optim. 84(4):869–912.CrossrefGoogle Scholar
  • Beale E, Forrest JJ (1976) Global optimization using special ordered sets. Math. Programming 10(1):52–69.CrossrefGoogle Scholar
  • Benson HP (1990) Separable concave minimization via partial outer approximation and branch and bound. Oper. Res. Lett. 9(6):389–394.CrossrefGoogle Scholar
  • Burer S, Letchford AN (2012) Non-convex mixed-integer nonlinear programming: A survey. Surveys Oper. Res. Management Sci. 17(2):97–106.CrossrefGoogle Scholar
  • Burlacu R, Geißler B, Schewe L (2020) Solving mixed-integer nonlinear programmes using adaptively refined mixed-integer linear programmes. Optim. Methods Software 35(1):37–64.CrossrefGoogle Scholar
  • Casado LG, Martínez JA, García I, Sergeyev YD (2003) New interval analysis support functions using gradient information in a global minimization algorithm. J. Global Optim. 25(4):345–362. CrossrefGoogle Scholar
  • Cormen TH, Leiserson CE, Rivest RL, Stein C (2009) Introduction to Algorithms (MIT Press, Cambridge, MA).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
  • Croxton KL, Gendron B, Magnanti TL (2007) Variable disaggregation in network flow problems with piecewise linear costs. Oper. Res. 55(1):146–157.LinkGoogle Scholar
  • D’Ambrosio C, Lee J, Skipper D, Thomopulos D (2020) Handling separable non-convexities using disjunctive cuts. Baiou, M Gendron B, Gunluk O, Mahjoub A, eds. Combinatorial Optimization: ISCO 2020, Lecture Notes in Computer Science, vol. 12176 (Springer, Cham, Switzerland), 102–114.CrossrefGoogle Scholar
  • Dantzig GB (1960) On the significance of solving linear programming problems with some integer variables. Econometrica 28(1):30–44.CrossrefGoogle Scholar
  • de Farias IR Jr, Zhao M, 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 IR, Kozyreff E, Gupta R, Zhao M (2013) Branch-and-cut for separable piecewise linear optimization and intersection with semi-continuous constraints. Math. Programming Comput. 5(1):75–112.CrossrefGoogle Scholar
  • Dey SS, Gupte A (2015) Analysis of MILP techniques for the pooling problem. Oper. Res. 63(2):412–427.LinkGoogle Scholar
  • Duguet A, Ngueveu SU (2022) Piecewise linearization of bivariate nonlinear functions: Minimizing the number of pieces under a bounded approximation error. Ljubíc I, Barahona F, Dey SS, Mahjoub AR, eds. Combinatorial Optimization: ISCO 2022, Lecture Notes in Computer Science, vol. 13526 (Springer, Cham, Switzerland), 117–129.CrossrefGoogle Scholar
  • Falk JE, Soland RM (1969) An algorithm for separable nonconvex programming problems. Management Sci. 15(9):550–569.LinkGoogle Scholar
  • Feijoo B, Meyer R (1988) Piecewise-linear approximation methods for nonseparable convex optimization. Management Sci. 34(3):411–419.LinkGoogle Scholar
  • Fourer R (1985) A simplex algorithm for piecewise-linear programming I: Derivation and proof. Math. Programming 33(2):204–233.CrossrefGoogle Scholar
  • Frenzen CL, Sasao T, Butler JT (2010) On the number of segments needed in a piecewise linear approximation. J. Comput. Appl. Math. 234(2):437–446.CrossrefGoogle Scholar
  • Füllner C, Rebennack S (2022) Non-convex nested Benders decomposition. Math. Programming 196(1):987–1024.CrossrefGoogle Scholar
  • Geisler B, Martin A, Morsi A, Schewe L (2012) Using piecewise linear functions for solving MINLPs. Lee J, Leyffer S, eds. Mixed Integer Nonlinear Programming, IMA Volumes in Mathematics and Its Applications, vol. 154 (Springer, Cham, Switzerland), 287–314.CrossrefGoogle Scholar
  • Ghaddar B, Gómez-Casares I, González-Díaz J, González-Rodríguez B, Pateiro-López B, Rodríguez-Ballesteros S (2023) Learning for spatial branching: An algorithm selection approach. INFORMS J. Comput. 35(5):1024–1043.LinkGoogle Scholar
  • Gorissen BL (2022) Interior point methods can exploit structure of convex piecewise linear functions with application in radiation therapy. SIAM J. Optim. 32(1):256–275.CrossrefGoogle Scholar
  • Gounaris CE, Floudas CA (2008) Tight convex underestimators for C2-continuous functions: I. Univariate functions. J. Global Optim. 42(1):51–67.CrossrefGoogle Scholar
  • Graham RL (1972) An efficient algorithm for determining the convex hull of a finite planar set. Inform. Processing Lett. 1(4):132–133.CrossrefGoogle Scholar
  • Grimstad B, Knudsen BR (2020) Mathematical programming formulations for piecewise polynomial functions. J. Global Optim. 77(3):455–486.CrossrefGoogle Scholar
  • Gupte A, Koster AM, Kuhnke S (2022) An adaptive refinement algorithm for discretizations of nonconvex QCQP. Schulz C, Ucar B, eds. 20th International Symposium on Experimental Algorithms: SEA 2022, Leibniz International Proceedings in Informatics (LIPIcs), vol. 233 (Schloss Dagstuhl Publishing, Wadern, Germany), 24:1–24:14.Google Scholar
  • Gurobi Optimization (2024) Documentation—General constraints. https://www.gurobi.com/documentation/current/refman/general_constraints.html.Google Scholar
  • Horst R (1986) A general class of branch-and-bound methods in global optimization with some new approaches for concave minimization. J. Optim. Theory Appl. 51(2):271–291.CrossrefGoogle Scholar
  • Hübner T, Gupte A, Rebennack S (2025) Spatial branch and bound for nonconvex separable piecewise linear optimization. https://github.com/INFORMSJoC/2024.0755.Google Scholar
  • Huchette J, Vielma JP (2023) Nonconvex piecewise linear functions: Advanced formulations and simple modeling tools. Oper. Res. 71(5):1835–1856.LinkGoogle Scholar
  • Keha AB, de Farias IR, Nemhauser GL (2004) Models for representing piecewise linear cost functions. Oper. Res. Lett. 32(1):44–48.CrossrefGoogle Scholar
  • Keha AB, de Farias IR, Nemhauser GL (2006) A branch-and-cut algorithm without binary variables for nonconvex piecewise linear optimization. Oper. Res. 54(5):847–858.LinkGoogle Scholar
  • Kesavan P, Allgor RJ, Gatzke EP, Barton PI (2004) Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs. Math. Programming 100(3):517–535.CrossrefGoogle Scholar
  • Kim J, Richard J-PP, Tawarmalani M (2024) Piecewise polyhedral relaxations of multilinear optimization. SIAM J. Optim. 34(4):3167–3193.Google Scholar
  • Kong L, Maravelias CT (2020) On the derivation of continuous piecewise linear approximating functions. INFORMS J. Comput. 32(3):531–546.LinkGoogle Scholar
  • Kontogiorgis S (2000) Practical piecewise-linear approximation for monotropic optimization. INFORMS J. Comput. 12(4):324–340.LinkGoogle Scholar
  • Leyffer S, Sartenaer A, Wanufelle E (2008) Branch-and-refine for mixed-integer nonconvex global optimization. Working Paper No. ANL/MCS-P1547-0908, Mathematics and Computer Science Division, Argonne National Laboratory, Lemont, IL.Google Scholar
  • Locatelli M, Schoen F (2013) Global Optimization: Theory, Algorithms, and Applications, MOS-SIAM Series on Optimization, vol. MO15 (SIAM, Philadelphia).CrossrefGoogle Scholar
  • Lyu B, Hicks IV, Huchette J (2025) Building formulations for piecewise linear relaxations of nonlinear functions. Oper. Res., ePub ahead of print April 24, https://doi.org/10.1287/opre.2023.0187.Google Scholar
  • Magnanti TL, Stratila D (2004) Separable concave optimization approximately equals piecewise linear optimization. Bienstock D, Nemhauser G, eds. Integer Programming and Combinatorial Optimization: IPCO 2004, Lecture Notes in Computer Science, vol. 3064 (Springer, Berlin), 234–243.CrossrefGoogle Scholar
  • Markowitz HM, Manne AS (1957) On the solution of discrete programming problems. Econometrica 25(1):84–110.CrossrefGoogle Scholar
  • Meyer RR (1976) Mixed integer minimization models for piecewise-linear functions of a single variable. Discrete Math. 16(2):163–171.CrossrefGoogle Scholar
  • Nagarajan H, Lu M, Wang S, Bent R, Sundar K (2019) An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs. J. Global Optim. 74(4):639–675.CrossrefGoogle Scholar
  • Natali JM, Pinto JM (2009) Piecewise polynomial interpolations and approximations of one-dimensional functions through mixed integer linear programming. Optim. Methods Software 24(4–5):783–803.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
  • Posypkin M, Usov A, Khamisov O (2020) Piecewise linear bounding functions in univariate global optimization. Soft Comput. 24(23):17631–17647.CrossrefGoogle Scholar
  • Rebennack S (2016) Computing tight bounds via piecewise linear functions through the example of circle cutting problems. Math. Methods Oper. Res. 84(1):3–57.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
  • Rebennack S, Nahapetyan A, Pardalos PM (2009) Bilinear modeling solution approach for fixed charge network flow problems. Optim. Lett. 3(3):347–355.CrossrefGoogle Scholar
  • Shectman JP, Sahinidis NV (1998) A finite algorithm for global minimization of separable concave programs. J. Global Optim. 12(1):1–36. CrossrefGoogle Scholar
  • Sherali HD (2001) On mixed-integer zero-one representations for separable lower-semicontinuous piecewise-linear functions. Oper. Res. Lett. 28(4):155–160.CrossrefGoogle Scholar
  • Soland RM (1971) An algorithm for separable nonconvex programming problems II: Nonconvex constraints. Management Sci. 17(11):759–773.LinkGoogle Scholar
  • Sundar K, Sanjeevi S, Nagarajan H (2022) Sequence of polyhedral relaxations for nonlinear univariate functions. Optim. Engrg. 23(2):877–894.CrossrefGoogle Scholar
  • Tawarmalani M, Sahinidis NV (2004) Global optimization of mixed-integer nonlinear programs: A theoretical and computational study. Math. Programming 99(3):563–591.CrossrefGoogle Scholar
  • Thakur LS (1978) Error analysis for convex separable programs: The piecewise linear approximation and the bounds on the optimal objective value. SIAM J. Appl. Math. 34(4):704–714.CrossrefGoogle Scholar
  • Toriello A, Vielma JP (2012) Fitting piecewise linear continuous functions. Eur. J. Oper. Res. 219(1):86–95.CrossrefGoogle Scholar
  • Tuy H (2016) Convex Analysis and Global Optimization, 2nd ed., Springer Optimization and Its Applications, vol. 110 (Springer, Cham, Switzerland).CrossrefGoogle Scholar
  • Tuy H, Horst R (1988) Convergence and restart in branch-and-bound algorithms for global optimization. Application to concave minimization and D.C. Optimization problems. Math. Programming 41(1):161–183.CrossrefGoogle Scholar
  • Vielma JP, Nemhauser GL (2011) Modeling disjunctive constraints with a logarithmic number of binary variables and constraints. Math. Programming 128(1–2):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
  • Vielma JP, Keha AB, Nemhauser GL (2008) Nonconvex, lower semicontinuous piecewise linear optimization. Discrete Optim. 5(2):467–488.CrossrefGoogle Scholar
  • Warwicker JA, Rebennack S (2022) A comparison of two mixed-integer linear programs for piecewise linear function fitting. INFORMS J. Comput. 34(2):1042–1047.LinkGoogle Scholar
  • Warwicker JA, Rebennack S (2024) Efficient continuous piecewise linear regression for linearising univariate non-linear functions. IISE Trans. 57(3):231–245.CrossrefGoogle Scholar
  • Wechsung A, Barton PI (2014) Global optimization of bounded factorable functions with discontinuities. J. Global Optim. 58(1):1–30.CrossrefGoogle Scholar
  • Yıldız S, Vielma JP (2013) Incremental and encoding formulations for mixed integer programming. Oper. Res. Lett. 41(6):654–658.CrossrefGoogle Scholar
  • Zhao M, de Farias IR Jr (2013) The piecewise linear optimization polytope: New inequalities and intersection with semi-continuous constraints. Math. Programming 141(1–2):217–255.CrossrefGoogle 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.