Abstract
The convex hull heuristic is a heuristic for mixed-integer programming problems with a nonlinear objective function and linear constraints. It is a matheuristic in two ways: it is based on the mathematical programming algorithm called simplicial decomposition, or SD (von Hohenbalken in Math Program 13:49–68, 1977), and at each iteration, one solves a mixed-integer programming problem with a linear objective function and the original constraints, and a continuous problem with a nonlinear objective function and a single linear constraint. Its purpose is to produce quickly feasible and often near optimal or optimal solutions for convex and nonconvex problems. It is usually multi-start. We have tested it on a number of hard quadratic 0–1 optimization problems and present numerical results for generalized quadratic assignment problems, cross-dock door assignment problems, quadratic assignment problems and quadratic knapsack problems. We compare solution quality and solution times with results from the literature, when possible.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahlatcioglu, A., Guignard, M.: The convex hull relaxation for nonlinear integer programs with linear constraints. Technical report, University of Pennsylvania, The Wharton School, OPIM Department, Philadelphia, PA (2007–2009)
Ahlatcioglu, A., Bussieck, M., Esen, M., Guignard, M., Jagla, J.-H., Meeraus, A.: Combining QCR and CHR for convex quadratic pure 0–1 programming problems with linear constraints. Ann. Oper. Res. 199, 33–49 (2012)
Ahn, S.: On solving some optimization problems in stochastic and integer programming with applications in finance and banking. Ph.D. dissertation, University of Pennsylvania (1997)
Albornoz, V.: Diseno de Modelos y Algoritmos de Optimizacion Robusta y su Aplicacion a la Planicacion Agregada de la Produccion. Ph.D. thesis, Universidad Catolica de Chile, Santiago, Chile (1998)
Anstreicher, K.M., Brixius, N.W.: Solving quadratic assignment problems using convex quadratic programming relaxations. Optim. Methods Softw. 16, 49–68 (2001)
Caprara, A., Pisinger, D., Toth, P.: Exact solution of the Quadratic Knapsack Problem. INFORMS J. Comput. 11, 125–137 (1999)
Cardoso da Silva, D.: Uma heuristica hibrida de busca em vizinhanca larga para o problema de atribuicao de portas de cross-dock. Universidade Federal Fluminense, Escola de Enghenaria, Mestrado em Engenharia de Producao (2013)
Contesse, L., Guignard M.: An augmented Lagrangean relaxation for nonlinear integer programming solved by the method of multipliers, Part I, theory and algorithms. Working Paper, OPIM Department, University of Pennsylvania (1995, latest revision 2009)
Cordeau, J.-F., Gaudioso, M., Laporte, G., Moccia, L.: A memetic heuristic for the generalized assignment problem. INFORMS J. Comput. 18, 433–443 (2006)
Frank, M., Wolfe, P.: An algorithm for quadratic programming. Naval Res. Q. 3, 95–109 (1956)
Geoffrion, A.M.: Lagrangean relaxation for integer programming. Math. Program. Stud. 2, 82–114 (1974)
Guignard, M.: Primal relaxations for integer programming. Invited plenary talk, VII CLAIO, Santiago, Chile (July 1994). Also Technical Report 94–02–01, University of Pennsylvania, OPIM Department (1994)
Guignard, M.: Lagrangean relaxation. TOP 11(2), 151–228 (2003)
Guignard, M.: A new, solvable, primal relaxation for nonlinear integer programming problems with linear constraints. Optimization Online (2007). http://www.optimizationonline.org/DB_FILE/2011/01/2904.pdf
Guignard, M., Hahn, P.M., Pessoa, A.A., Cardoso da Silva, D.: Algorithms for the cross-dock door assignment problem. In: Proceedings of the 4th International Workshop on Model-Based Metaheuristics. Angra dos Reis, Brazil, pp. 1–12 (2012)
Hearn, D.W., Lawphongpanich, S., Ventura, J.A.: Finiteness in restricted simplicial decomposition. Oper. Res. Lett. 4, 125–130 (1985)
Lee, C.-G., Ma, Z.: The generalized quadratic assignment problem. Research Report, Department of Mechanical and Industrial Engineering, University of Toronto (2003)
Létocart, L., Nagih A., Plateau G.: Reoptimization in Lagrangian methods for the 01 quadratic knapsack problem. Computers and O.R., Special Issue on Knapsack Problems and Applications, 39, 12–18 (2012)
Létocart, L., Plateau, G.: Private communication (2015–2016)
Michelon, P., Maculan, N.: Solving the Lagrangean dual problem in integer programming. Report 822, Departement d’ Informatique et de Recherche Operationnelle, University of Montreal (1992)
Patriksson, M.: Simplicial decomposition algorithms. In: Floudas, C., Pardalos, P. (eds.) Encyclopedia of Optimization. Springer, Boston (2008)
Poljak, S., Rendl, F., Wolkowicz, H.: A recipe for best semidefinite relaxation for 0–1 quadratic programming. J. Glob. Optim. 7, 51–73 (1995)
Resende, M.G.C., Ramakrishnan, K.G., Drezner, Z.: Computing lower bounds for the quadratic assignment problem with an interior point algorithm for linear programming. Oper. Res. 43, 781–791 (1995)
Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero–one programming problems. SIAM J. Discrete Math. 3, 411–430 (1990)
von Hohenbalken, B.: Simplicial decomposition in nonlinear programming algorithms. Math. Program. 13, 49–68 (1977)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Guignard, M., Ahlatcioglu, A. The convex hull heuristic for nonlinear integer programming problems with linear constraints and application to quadratic 0–1 problems. J Heuristics 27, 251–265 (2021). https://doi.org/10.1007/s10732-019-09433-w
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10732-019-09433-w