Abstract
In the context of convex mixed integer nonlinear programming (MINLP), we investigate how the outer approximation method and the generalized Benders decomposition method are affected when the respective nonlinear programming (NLP) subproblems are solved inexactly. We show that the cuts in the corresponding master problems can be changed to incorporate the inexact residuals, still rendering equivalence and finiteness in the limit case. Some numerical results will be presented to illustrate the behavior of the methods under NLP subproblem inexactness.
Similar content being viewed by others
References
CMU/IBM MINLP Project. http://egon.cheme.cmu.edu/ibm/page.htm
IBM ILOG CPLEX. http://www-01.ibm.com/software/integration/optimization/cplex-optimizer. Version 12.4, 2012
Ahlatçıoğlu, A., Guignard, M.: Convex hull relaxation (CHR) for convex and nonconvex MINLP problems with linear constraints (2011)
Belotti P., Lee J., Liberti L., Margot F., Wächter A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24, 597–634 (2009)
Benders J.F.: Partitioning procedures for solving mixed-variables programming problems. Numer. Math. 4, 238–252 (1962)
Bonami P., Biegler L.T., Conn A.R., Cornuéjols G., Grossmann I.E., Laird C.D., Lee J., Lodi A., Margot F., Sawaya N., Wächter A.: An algorithmic framework for convex mixed integer nonlinear programs. Discret. Optim. 5, 186–204 (2008)
Bonami P., Cornuéjols G., Lodi A., Margot F.: A feasibility pump for mixed integer nonlinear programs. Math. Program. 119, 331–352 (2009)
Bonami P., Gonçalves J.P.M.: Heuristics for convex mixed integer nonlinear programs. Comput. Optim. Appl. 51, 729–747 (2012)
Castillo I., Westerlund J., Emet S., Westerlund T.: Optimization of block layout design problems with unequal areas: A comparison of MILP and MINLP optimization methods. Comput. Chem. Eng. 30, 54–69 (2005)
Duran M.A., Grossmann I.E.: An out-approximation algorithm for a class of mixed-integer nonlinear programs. Math. Program. 36, 307–339 (1986)
Fletcher R., Leyffer S.: Solving mixed integer nonlinear programs by outer approximation. Math. Program. 66, 327–349 (1994)
Flippo O.E., Rinnooy Kan A.H.G.: Decomposition in general mathematical programming. Math. Program. 60, 361–382 (1993)
Geoffrion A.M.: Generalized Benders decomposition. J. Optim. Theory Appl. 10, 237–260 (1972)
Grossmann I.E.: Review of nonlinear mixed-integer and disjunctive programming techniques. Optim. Eng. 3, 227–252 (2002)
Kelley J.E.: The cutting-plane method for solving convex programs. J. Soc. Ind. Appl. Math. 8, 703–712 (1960)
Leyffer, S.: MacMINLP. http://wiki.mcs.anl.gov/leyffer/index.php/MacMINLP
Nannicini, G., Belotti, P., Lee, J., Linderoth, J., Margot, F., Wächter, A.: A probing algorithm for MINLPs with early detection of failures by SVM. In: CPAIOR 2011: The 8th International Conference on Integration of Artificial Intelligence and Operations Research, volume 6697 of Lecture Notes in Computer Science, pp. 154–169. Springer, New York (2011)
Quesada I., Grossmann I.E.: An LP/NLP based branch and bound algorithm for convex MINLP optimization problems. Comput. Chem. Eng. 16, 937–947 (1992)
Tawarmalani M., Sahinidis N.V.: Global optimization of mixed-integer nonlinear programs: A theoretical and computational study. Math. Program. 99, 563–591 (2004)
Westerlund T., Pettersson F.: An extended cutting plane method for solving convex MINLP problems. Comput. Chem. Eng. 19, 131–136 (1995)
Westerlund T., Pörn R.: Solving pseudo-convex mixed integer optimization problems by cutting plane techniques. Optim. Eng. 3, 253–280 (2002)
Westerlund T., Skrifvars H., Harjunkoski I., Pörn R.: An extended cutting plane method for a class of non-convex MINLP problems. Comput. Chem. Eng. 22, 357–365 (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
M. Li was supported by FCT under the scholarship SFRH/BD/33369/2008.
L. N. Vicente was supported by FCT under the grant PTDC/MAT/098214/2008.
Rights and permissions
About this article
Cite this article
Li, M., Vicente, L.N. Inexact solution of NLP subproblems in MINLP. J Glob Optim 55, 877–899 (2013). https://doi.org/10.1007/s10898-012-0010-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-012-0010-5