Abstract
A novel method, entitled the discrete global descent method, is developed in this paper to solve discrete global optimization problems and nonlinear integer programming problems. This method moves from one discrete minimizer of the objective function f to another better one at each iteration with the help of an auxiliary function, entitled the discrete global descent function. The discrete global descent function guarantees that its discrete minimizers coincide with the better discrete minimizers of f under some standard assumptions. This property also ensures that a better discrete minimizer of f can be found by some classical local search methods. Numerical experiments on several test problems with up to 100 integer variables and up to 1.38 × 10104 feasible points have demonstrated the applicability and efficiency of the proposed method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Borghi C.A., Casadei D., Febbri M., Serra G. (1998). Reduction of the torque ripple in permanent magnet actuators by a multi-objective minimization technique. IEEE Trans. Magnet. 34(5):2869–2872
Borghi C.A., Fabbri M. (1997). A combined technique for the global optimization of the inverse electromagnetic problem solution. IEEE Trans. Magnet. 33(2):1947–1950
Borghi C.A., Febbri M., Di Barba P., Savini A. (1999). A comparative study of loneys solenoid by different techniques of global optimization. Int. J. Appl. Electromagnet. Mech. 10(5):417–423
Conley W. (1980). Computer Optimization Techniques. Petrocelli Books Inc, New York
Fisher M.L. (1981). The Lagrangian relaxation method for solving integer programming problems. Manage. Sci. 27(1):1–18
Ge, R.-P.: (1990) A filled function method for finding a global minimizer of a function of several variables. Math. Program. 46(2), 191–204 (1990)
Ge R.-P., Huang C.-B. (1989). A continuous approach to nonlinear integer programming. Appl. Mathe. Comput. 34(1):39–60
Ge R.-P., Qin Y.-F. (1987). A class of filled functions for finding global minimizers of a function of several variables. J. Optim. Theory Appl. 54(2):241–252
Geoffirion A.M. (1974). Lagrangian relaxation for integer programming. Mathe. Program. Study 2:82–114
Goldstein A.A., Price J.F. (1971). On descent from local minima. Mathe. Comput. 25(115):569–574
Gupta O.K., Ravindran A. (1985). Branch and bound experiments in convex nonlinear integer programming. Manage. Scie 31(12):1533–1546
Hock W., Schittkowski K. (1981). Test Examples for Nonlinear Programming Codes. Springer-Verlag, New York
Körner F. (1988). A new branching rule for the branch and bound algorithm for solving nonlinear integer programming problems. BIT 28(3):701–708
Li D., Sun X.-L. (2000). Success guarantee of dual search in integer programming: pth power Lagrangian method. J. Global Optim. 18:235–254
Li D., Sun X. (2006). Nonlinear Integer Programming. Springer, Boston
Li D., White D.J. (2000). pth Power Lagrangian method for integer programming. Ann. Operat. Res. 98:151–170
Liu X. (2001). Finding global minima with a computable filled function. J. Global Optim. 19(2):151–161
Liu X., Xu W. (2004). A new filled function applied to global optimization. Comp. Operat. Res. 31(1):61–80
Mohan C., Nguyen H.T. (1999). A controlled random search technique incorporating the simulated annealing concept for solving integer and mixed integer global optimization problems. Comput. Optim. Appl. 14:103–132
Moré J.J., Garbow B.S., Hillstrom K.E. (1981). Testing unconstrained optimization software. ACM Trans. Math. Software 7(1):17–41
Ng, C.-K.: High performance continuous/discrete global optimization methods. Ph.D. Dissertation, Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Hong Kong S.A.R., P.R.China (2003)
Ng, C.-K., Li, D., Zhang, L.-S.: Filled function approaches to nonlinear integer programming: a survey. In: Hou, S.H., Yang, X.M., Chen, G.Y. (eds.) Frontiers in Optimization and Control. Kluwer Academic Publishers. To appear (2005a)
Ng C.-K., Zhang L.-S., Li D., Tian W.-W. (2005b). Discrete filled function method for discrete global optimization. Comput. Optim. Appl. 31(1):87–115
Schittkowski, K. More Test Examples for Nonlinear Programming Codes. Springer-Verlag (1987)
Sun X.-L., Li D. (2000). Asymptotic strong duality for bounded integer programming: a logarithmic-exponential dual formulation. Math. Operat. Res. 25:625–644
Xu Z., Huang H.-X., Pardalos P.M., Xu C.-X. (2001). Filled functions for unconstrained global optimization. J. Global Optim. 20(1):49–65
Yang X.Q., Goh C.J. (2001). A nonlinear Lagrangian function for discrete optimization problems. In: Migdalas A., Pardalos P.M., and Värbrand P. (eds) From Local to Global Optimization. Kluwer Academic Publishers, Dordrecht
Zhang L.-S., Gao F., Zhu W.-X. (1999). Nonlinear integer programming and global optimization. J. Comput. Math. 17(2):179–190
Zhang L.-S., Ng C.-K., Li D., Tian W.-W. (2004). A new filled function method for global optimization. J. Global Optim. 28(1):17–43
Zheng Q., Zhuang D.-M. (1995). Integral global minimization: algorithms, implementations and numerical tests. J. Global Optim 7(4):421–454
Zhu W.-X. (1998). An approximate algorithm for nonlinear integer programming. Appl. Math. Comput. 93(2–3):183–193
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ng, CK., Li, D. & Zhang, LS. Discrete global descent method for discrete global optimization and nonlinear integer programming. J Glob Optim 37, 357–379 (2007). https://doi.org/10.1007/s10898-006-9053-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-006-9053-9