Abstract
The paper researches a class of nonlinear integer programming problems the objective function of which is the sum of the products of some nonnegative linear functions in the given rectangle and the constraint functions of which are all linear as well as strategy variables of which are all integer ones. We give a linear programming relax-PSO hybrid bound algorithm for solving the problem. The lower bound of the optimal value of the problem is determined by solving a linear programming relax which is obtained through equally converting the objective function into the exponential-logarithmic composite function and linearly lower approximating each exponential function and each logarithmic function over the rectangles. The upper bound of the optimal value and the feasible solution of it are found and renewed with particle swarm optimization (PSO). It is shown by the numerical results that the linear programming relax-PSO hybrid bound algorithm is better than the branch-and-bound algorithm in the computational scale and the computational time and the computational precision and overcomes the convergent difficulty of PSO.
The work is supported by the Foundations of Post-doctoral Science in China (grants 2006041001) and National Natural Science in Ningxia (2006), and by the Science Research Projects of National Committee in China and the Science Research Project of Ningxia’s Colleges and Universities in 2005.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. John Wiley and sons, New York (1988)
Kuno, T.: Solving a class of multiplicative programs with 0-1 knapsack constraints. Journal of Optimization Theory and Applications 103, 121–125 (1999)
Barrientos, O., Correa, R., Reyes, P., Valdebenito, A.: A brand and bound method for solving integer separable concave problems. Computational Optimization and Applications 26, 155–171 (2003)
Horst, R., Tuy, H.: Global optimization, deterministic approaches. Springer, Heidelberg (1996)
Gao, Y.L., Xu, C.X, Wang, Y.J., Zhang, L.S.: A new two-level linear relaxed bound method for geometric programming problem. Applied Mathematics and Computation 164, 117–131 (2005)
Laughunn, D.J.: Quadratic binary programming with applications to capital-budgeting problem. Operations Research 14, 454–461 (1970)
Krarup, J., Pruzan, P.M.: Computer-aided layout design. Mathematical Programming Study 9, 75–94 (1978)
Markovitz, H.M.: Portfolio selection. Wily, New York (1978)
Witzgall, C.: Mathematical method of site selection for Electric Message Systems(EMS), NBS Internet Report (1975)
Rhys, J.: A selection problem of shared fixed costs on network flow. Management Science 17, 200–207 (1970)
Eberhart, R.C., Shi, Y.H.: Particle swarm optimization: development, applications and resources. In: Proceedings of the IEEE International Conference on Evolutionary Computation, pp. 81–86 (2002)
Laskari, E.C., Parsopoulos, K.E., Vrahatis, M.N.: Particle swarm optimization for integer programming. In: Proceedings of the IEEE International Conference on Evolutionary Computation, pp. 1582–1587 (1978)
Eberhart, R.C., Shi, Y.H.: Comparison between genetic algorithms and particle swarm optimization: development, applications and resources, Evolutionary Programming, pp. 611–615 (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gao, Y., Xu, C., Li, J. (2007). Linear Programming Relax-PSO Hybrid Bound Algorithm for a Class of Nonlinear Integer Programming Problems. In: Wang, Y., Cheung, Ym., Liu, H. (eds) Computational Intelligence and Security. CIS 2006. Lecture Notes in Computer Science(), vol 4456. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74377-4_4
Download citation
DOI: https://doi.org/10.1007/978-3-540-74377-4_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74376-7
Online ISBN: 978-3-540-74377-4
eBook Packages: Computer ScienceComputer Science (R0)