A Parametric Linearization Approach for Solving Zero-One Nonlinear Programming Problems

ABSTRACT

In this paper a new approach for obtaining an approximation global optimum solution of zero-one nonlinear programming (0-1 NP) problem which we call it Parametric Linearization Approach (P.L.A) is proposed. By using this approach the problem is transformed to a sequence of linear programming problems. The approximately solution of the original 0-1 NP problem is obtained based on the optimum values of the objective functions of this sequence of linear programming problems defined by (P.L.A).

In this paper a new approach for obtaining an approximation global optimum solution of zero-one nonlinear programming (0-1 NP) problem which we call it Parametric Linearization Approach (P.L.A) is proposed. By using this approach the problem is transformed to a sequence of linear programming problems. The approximately solution of the original 0-1 NP problem is obtained based on the optimum values of the objective functions of this sequence of linear programming problems defined by (P.L.A).

Cite this paper

nullA. Vaziri, A. Kamyad and S. Efatti, "A Parametric Linearization Approach for Solving Zero-One Nonlinear Programming Problems,"*Applied Mathematics*, Vol. 2 No. 10, 2011, pp. 1207-1212. doi: 10.4236/am.2011.210168.

nullA. Vaziri, A. Kamyad and S. Efatti, "A Parametric Linearization Approach for Solving Zero-One Nonlinear Programming Problems,"

References

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[2] J. Mitchell, P. M. Pardalos and M. G. C. Re-sende, “Interior Point Method for Combinatorial Optimi-zation,” In: D. Z. Du and P. Paradalos, Eds., Handbook of Combinatorial Optimization, Vol. 1, 1998, pp. 189-298.

[3] G. L. Nemhauser and L. A. Wolsey, “In-teger and Combinatorial Optimization,” John Wiley and Sons, New York, 1988.

[4] P. M. Paradalos, “Continuous Approaches to Discrete Optimization Problems,” In: G. Di Pillo and F. Gianessi, Eds., Nonlinear Optimization and Applications, Plenum Publishing, New York, 1996, pp. 313-328.

[5] M. S. Chern and R. H. Jan, “Reliability Optimization Problems with Multiple Constraints,” IEEE Transactions on Reliability, 1986, Vol. 35, No. 4, pp. 431-436. doi:10.1109/TR.1986.4335497

[6] K. B. Mira and U. Sharma, “An Efficient Algorithm to Solve Integer Pro-gramming Problems Arising in System-Reliability De-sign,” IEEE Transactions on Reliability, Vol. 40, No. 1, 1991, pp. 81-91.

[7] E. L. Lawlev and M. D. Bell, “A Method for Solving Discrete Optimization Problems,” Operations Research, Vol. 14, No. 6, 1966, pp. 1098-1112. doi:10.1287/opre.14.6.1098

[8] P. C. Gilmol and R. E. Gomory, “The Theory and Computation of Knapsack Function,” Operations Research, Vol. 14, 1966, pp. 1045-1074. doi:10.1287/opre.14.6.1045

[9] E. Balas, “An Additive Algorithm for Solving Linear Programs with 0-1 Varia-ble,” Operations Research, Vol. 13, No. 4, 1965, pp. 517-545. doi:10.1287/opre.13.4.517

[10] F. Glover, “Improved Linear Integer Programming Formulations of Nonlinear Integer Problems,” Management Science, Vol. 22, No. 4, 1975, pp. 455-460. doi:10.1287/mnsc.22.4.455

[11] M. Oral and O. Kettani, “Reformulating Nonlinear Combinatorial Optimization Problems for Higher Computational Efficiency,” European Journal of Operational Research, Vol. 58, No. 2, 1992, pp. 236-249. doi:10.1016/0377-2217(92)90210-Z

[12] M. Oral and O. Kettani, “A Linearization Procedure for Quadratic and Cubic Mixed Integer Problems,” Opera-tions Research, Vol. 40, Supplement 1, 1992, pp. s109- s116.

[13] P. Hansen and C. Meyer, “Improved Compact Linearizations for the Unconstrained Quadratic 0-1 Minimization Problem,” Discrete Applied Mathematics, Vol. 157, 2009, pp. 1267-1290. doi:10.1016/j.dam.2007.12.008

[1] P. Hansen, “Method of Nonlinear 0-1 Programming,” Annals of Discrete Mathematics, Vol. 5, 1979, pp. 53-70.

[2] J. Mitchell, P. M. Pardalos and M. G. C. Re-sende, “Interior Point Method for Combinatorial Optimi-zation,” In: D. Z. Du and P. Paradalos, Eds., Handbook of Combinatorial Optimization, Vol. 1, 1998, pp. 189-298.

[3] G. L. Nemhauser and L. A. Wolsey, “In-teger and Combinatorial Optimization,” John Wiley and Sons, New York, 1988.

[4] P. M. Paradalos, “Continuous Approaches to Discrete Optimization Problems,” In: G. Di Pillo and F. Gianessi, Eds., Nonlinear Optimization and Applications, Plenum Publishing, New York, 1996, pp. 313-328.

[5] M. S. Chern and R. H. Jan, “Reliability Optimization Problems with Multiple Constraints,” IEEE Transactions on Reliability, 1986, Vol. 35, No. 4, pp. 431-436. doi:10.1109/TR.1986.4335497

[6] K. B. Mira and U. Sharma, “An Efficient Algorithm to Solve Integer Pro-gramming Problems Arising in System-Reliability De-sign,” IEEE Transactions on Reliability, Vol. 40, No. 1, 1991, pp. 81-91.

[7] E. L. Lawlev and M. D. Bell, “A Method for Solving Discrete Optimization Problems,” Operations Research, Vol. 14, No. 6, 1966, pp. 1098-1112. doi:10.1287/opre.14.6.1098

[8] P. C. Gilmol and R. E. Gomory, “The Theory and Computation of Knapsack Function,” Operations Research, Vol. 14, 1966, pp. 1045-1074. doi:10.1287/opre.14.6.1045

[9] E. Balas, “An Additive Algorithm for Solving Linear Programs with 0-1 Varia-ble,” Operations Research, Vol. 13, No. 4, 1965, pp. 517-545. doi:10.1287/opre.13.4.517

[10] F. Glover, “Improved Linear Integer Programming Formulations of Nonlinear Integer Problems,” Management Science, Vol. 22, No. 4, 1975, pp. 455-460. doi:10.1287/mnsc.22.4.455

[11] M. Oral and O. Kettani, “Reformulating Nonlinear Combinatorial Optimization Problems for Higher Computational Efficiency,” European Journal of Operational Research, Vol. 58, No. 2, 1992, pp. 236-249. doi:10.1016/0377-2217(92)90210-Z

[12] M. Oral and O. Kettani, “A Linearization Procedure for Quadratic and Cubic Mixed Integer Problems,” Opera-tions Research, Vol. 40, Supplement 1, 1992, pp. s109- s116.

[13] P. Hansen and C. Meyer, “Improved Compact Linearizations for the Unconstrained Quadratic 0-1 Minimization Problem,” Discrete Applied Mathematics, Vol. 157, 2009, pp. 1267-1290. doi:10.1016/j.dam.2007.12.008