Spatial Branch-and-Bound for Nonconvex Separable Piecewise Linear Optimization
Published Online:27 Jun 2025https://doi.org/10.1287/ijoc.2024.0755
References
- (2005) Branching rules revisited. Oper. Res. Lett. 33(1):42–54.Crossref, Google Scholar
- (2019) Error bounds for monomial convexification in polynomial optimization. Math. Programming 175(1–2):355–393.Crossref, Google Scholar
- (1998) A global optimization method, αBB, for general twice-differentiable constrained NLPs—I. Theoretical advances. Comput. Chemical Engrg. 22(9):1137–1158.Crossref, Google Scholar
- (2000) On finitely terminating branch-and-bound algorithms for some global optimization problems. SIAM J. Optim. 10(4):1049–1057.Crossref, Google Scholar
- (1996) The quickhull algorithm for convex hulls. ACM Trans. Math. Software 22(4):469–483.Crossref, Google Scholar
- (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.Crossref, Google Scholar
- (2022) Compact mixed-integer programming formulations in quadratic optimization. J. Global Optim. 84(4):869–912.Crossref, Google Scholar
- (1976) Global optimization using special ordered sets. Math. Programming 10(1):52–69.Crossref, Google Scholar
- (1990) Separable concave minimization via partial outer approximation and branch and bound. Oper. Res. Lett. 9(6):389–394.Crossref, Google Scholar
- (2012) Non-convex mixed-integer nonlinear programming: A survey. Surveys Oper. Res. Management Sci. 17(2):97–106.Crossref, Google Scholar
- (2020) Solving mixed-integer nonlinear programmes using adaptively refined mixed-integer linear programmes. Optim. Methods Software 35(1):37–64.Crossref, Google Scholar
- (2003) New interval analysis support functions using gradient information in a global minimization algorithm. J. Global Optim. 25(4):345–362. Crossref, Google Scholar
- (2009) Introduction to Algorithms (MIT Press, Cambridge, MA).Google Scholar
- (2003) A comparison of mixed-integer programming models for nonconvex piecewise linear cost minimization problems. Management Sci. 49(9):1268–1273.Link, Google Scholar
- (2007) Variable disaggregation in network flow problems with piecewise linear costs. Oper. Res. 55(1):146–157.Link, Google Scholar
- (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.Crossref, Google Scholar
- (1960) On the significance of solving linear programming problems with some integer variables. Econometrica 28(1):30–44.Crossref, Google Scholar
- (2008) A special ordered set approach for optimizing a discontinuous separable piecewise linear function. Oper. Res. Lett. 36(2):234–238.Crossref, Google Scholar
- (2013) Branch-and-cut for separable piecewise linear optimization and intersection with semi-continuous constraints. Math. Programming Comput. 5(1):75–112.Crossref, Google Scholar
- (2015) Analysis of MILP techniques for the pooling problem. Oper. Res. 63(2):412–427.Link, Google Scholar
- (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.Crossref, Google Scholar
- (1969) An algorithm for separable nonconvex programming problems. Management Sci. 15(9):550–569.Link, Google Scholar
- (1988) Piecewise-linear approximation methods for nonseparable convex optimization. Management Sci. 34(3):411–419.Link, Google Scholar
- (1985) A simplex algorithm for piecewise-linear programming I: Derivation and proof. Math. Programming 33(2):204–233.Crossref, Google Scholar
- (2010) On the number of segments needed in a piecewise linear approximation. J. Comput. Appl. Math. 234(2):437–446.Crossref, Google Scholar
- (2022) Non-convex nested Benders decomposition. Math. Programming 196(1):987–1024.Crossref, Google Scholar
- (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.Crossref, Google Scholar
- (2023) Learning for spatial branching: An algorithm selection approach. INFORMS J. Comput. 35(5):1024–1043.Link, Google Scholar
- (2022) Interior point methods can exploit structure of convex piecewise linear functions with application in radiation therapy. SIAM J. Optim. 32(1):256–275.Crossref, Google Scholar
- (2008) Tight convex underestimators for C2-continuous functions: I. Univariate functions. J. Global Optim. 42(1):51–67.Crossref, Google Scholar
- (1972) An efficient algorithm for determining the convex hull of a finite planar set. Inform. Processing Lett. 1(4):132–133.Crossref, Google Scholar
- (2020) Mathematical programming formulations for piecewise polynomial functions. J. Global Optim. 77(3):455–486.Crossref, Google Scholar
- (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
- (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.Crossref, Google Scholar
- (2025) Spatial branch and bound for nonconvex separable piecewise linear optimization. https://github.com/INFORMSJoC/2024.0755.Google Scholar
- (2023) Nonconvex piecewise linear functions: Advanced formulations and simple modeling tools. Oper. Res. 71(5):1835–1856.Link, Google Scholar
- (2004) Models for representing piecewise linear cost functions. Oper. Res. Lett. 32(1):44–48.Crossref, Google Scholar
- (2006) A branch-and-cut algorithm without binary variables for nonconvex piecewise linear optimization. Oper. Res. 54(5):847–858.Link, Google Scholar
- (2004) Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs. Math. Programming 100(3):517–535.Crossref, Google Scholar
- (2024) Piecewise polyhedral relaxations of multilinear optimization. SIAM J. Optim. 34(4):3167–3193.Google Scholar
- (2020) On the derivation of continuous piecewise linear approximating functions. INFORMS J. Comput. 32(3):531–546.Link, Google Scholar
- (2000) Practical piecewise-linear approximation for monotropic optimization. INFORMS J. Comput. 12(4):324–340.Link, Google Scholar
- (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
- (2013) Global Optimization: Theory, Algorithms, and Applications, MOS-SIAM Series on Optimization, vol. MO15 (SIAM, Philadelphia).Crossref, Google Scholar
- (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
- (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.Crossref, Google Scholar
- (1957) On the solution of discrete programming problems. Econometrica 25(1):84–110.Crossref, Google Scholar
- (1976) Mixed integer minimization models for piecewise-linear functions of a single variable. Discrete Math. 16(2):163–171.Crossref, Google Scholar
- (2019) An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs. J. Global Optim. 74(4):639–675.Crossref, Google Scholar
- (2009) Piecewise polynomial interpolations and approximations of one-dimensional functions through mixed integer linear programming. Optim. Methods Software 24(4–5):783–803.Crossref, Google Scholar
- (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.Crossref, Google Scholar
- (2020) Piecewise linear bounding functions in univariate global optimization. Soft Comput. 24(23):17631–17647.Crossref, Google Scholar
- (2016) Computing tight bounds via piecewise linear functions through the example of circle cutting problems. Math. Methods Oper. Res. 84(1):3–57.Crossref, Google Scholar
- (2015) Continuous piecewise linear delta-approximations for univariate functions: Computing minimal breakpoint systems. J. Optim. Theory Appl. 167(2):617–643.Crossref, Google Scholar
- (2020) Piecewise linear function fitting via mixed-integer linear programming. INFORMS J. Comput. 32(2):507–530.Link, Google Scholar
- (2009) Bilinear modeling solution approach for fixed charge network flow problems. Optim. Lett. 3(3):347–355.Crossref, Google Scholar
- (1998) A finite algorithm for global minimization of separable concave programs. J. Global Optim. 12(1):1–36. Crossref, Google Scholar
- (2001) On mixed-integer zero-one representations for separable lower-semicontinuous piecewise-linear functions. Oper. Res. Lett. 28(4):155–160.Crossref, Google Scholar
- (1971) An algorithm for separable nonconvex programming problems II: Nonconvex constraints. Management Sci. 17(11):759–773.Link, Google Scholar
- (2022) Sequence of polyhedral relaxations for nonlinear univariate functions. Optim. Engrg. 23(2):877–894.Crossref, Google Scholar
- (2004) Global optimization of mixed-integer nonlinear programs: A theoretical and computational study. Math. Programming 99(3):563–591.Crossref, Google Scholar
- (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.Crossref, Google Scholar
- (2012) Fitting piecewise linear continuous functions. Eur. J. Oper. Res. 219(1):86–95.Crossref, Google Scholar
- (2016) Convex Analysis and Global Optimization, 2nd ed., Springer Optimization and Its Applications, vol. 110 (Springer, Cham, Switzerland).Crossref, Google Scholar
- (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.Crossref, Google Scholar
- (2011) Modeling disjunctive constraints with a logarithmic number of binary variables and constraints. Math. Programming 128(1–2):49–72.Crossref, Google Scholar
- (2010) Mixed-integer models for nonseparable piecewise-linear optimization: Unifying framework and extensions. Oper. Res. 58(2):303–315.Link, Google Scholar
- (2008) Nonconvex, lower semicontinuous piecewise linear optimization. Discrete Optim. 5(2):467–488.Crossref, Google Scholar
- (2022) A comparison of two mixed-integer linear programs for piecewise linear function fitting. INFORMS J. Comput. 34(2):1042–1047.Link, Google Scholar
- (2024) Efficient continuous piecewise linear regression for linearising univariate non-linear functions. IISE Trans. 57(3):231–245.Crossref, Google Scholar
- (2014) Global optimization of bounded factorable functions with discontinuities. J. Global Optim. 58(1):1–30.Crossref, Google Scholar
- (2013) Incremental and encoding formulations for mixed integer programming. Oper. Res. Lett. 41(6):654–658.Crossref, Google Scholar
- (2013) The piecewise linear optimization polytope: New inequalities and intersection with semi-continuous constraints. Math. Programming 141(1–2):217–255.Crossref, Google Scholar

