Abstract
This paper was motivated by an industrial optimization problem arisen at the Erdenet Mining Corporation (Mongolia). The problem involved real industrial data turned out to be a quadratically constrained quadratic programming problem, which we solve by applying the global search theory for general DC programming. According to the theory, first, we obtain an explicit DC representation of the nonconvex functions involved in the problem. Second, we perform a local search that takes into account the structure of the problem in question. Further, we construct procedures for escaping critical points provided by the local search method. In particular, we propose a new way of constructing an approximation of the level set based on conjugated vectors. The computational simulation demonstrates that the proposed method is a quite flexible tool which can fast provide operations staff with good solutions to achieve the best performance according to specific requirements.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Barrients, O., Correa, R.: An algorithm for global minimizatoion of linearly constrained quadratic functions. J. Global Optim. 16, 77–93 (2000). https://doi.org/10.1023/A:1008306625093
Boer, E.P.J., Hendrih, E.M.T.: Global optimization problems in optimal design of experiments in regression models. J. Global Optim. 18, 385–398 (2000). https://doi.org/10.1023/A:1026552318150
Bomze, I., Danninger, G.: A finite algorithm for solving general quadratic problem. J. Global Optim. 4(1), 1–16 (1994). https://doi.org/10.1007/BF01096531
Bonnans, J.-F., Gilbert, J.C., Lemaréchal, C., Sagastizábal, C.A.: Numerical Optimization: Theoretical and Practical Aspects, 2nd edn. Springer-Verlag, Heidelberg (2006). https://doi.org/10.1007/978-3-540-35447-5
Enkhbat, R., Gruzdeva, T.V., Barkova, M.V.: D.C. programming approach for solving an applied ore-processing problem. J. Ind. Manag. Optim. 14(2), 613–623 (2018)
Fedorov, V.V.: Theory of Optimal Experiments. Academic Press, New-York (1972)
Gruzdeva, T.V., Strekalovsky, A.S.: Local Search in Problems with Nonconvex Constraints. Comput. Math. Math. Phys. 47, 381–396 (2007). https://doi.org/10.1134/S0965542507030049
Gruzdeva, T., Strekalovsky, A.: An approach to fractional programming via D.C. constraints problem: local search. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) DOOR 2016. LNCS, vol. 9869, pp. 404–417. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-44914-2_32
Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. Springer-Verlag, Heidelberg (1993). https://doi.org/10.1007/978-3-662-02796-7
Horst, R., Pardalos, P., Thoai, N.V.: Introduction to Global Optimization. Kluwer Academic Publishers, Dordrecht (1995)
Horst, R., Pardalos, P.M.: Handbook of Global Optimization. Kluwer Academic, Dordrecht (1995)
Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (2006). https://doi.org/10.1007/978-0-387-40065-5
Pardalos, P.M., Rosen, J.B. (eds.): Constrained Global Optimization: Algorithms and Applications. LNCS, vol. 268. Springer, Heidelberg (1987). https://doi.org/10.1007/BFb0000035
Pardalos, P.M., Schnitger, J.: Checking local optimality in constrained quadratic programming is NP-hard. Oper. Res. Lett. 7, 33–35 (1988)
Strekalovsky, A.S.: On the merit and penalty functions for the D.C. Optimization. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) DOOR 2016. LNCS, vol. 9869, pp. 452–466. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-44914-2_36
Strekalovsky, A.S.: On local search in d.c. optimization problems. Appl. Math. Comput. 255, 73–83 (2015)
Strekalovsky, A.S.: On solving optimization problems with hidden nonconvex structures. In: Rassias, T.M., Floudas, C.A., Butenko, S. (eds.) Optimization in Science and Engineering, pp. 465–502. Springer, New York (2014). https://doi.org/10.1007/978-1-4939-0808-0_23
Strekalovsky, A.S.: Elements of Nonconvex Optimization. Nauka, Novosibirsk (2003). (in Russian)
Strekalovsky, A.S.: Global optimality conditions and exact penalization. Optim. Lett. 13(3), 597–615 (2019). https://doi.org/10.1007/s11590-017-1214-x
Strekalovsky, A.S., Yakovleva, T.V.: On a local and global search involved in nonconvex optimization problems. Autom. Remote Control. 65, 375–387 (2004). https://doi.org/10.1023/B:AURC.0000019368.45522.7a
Tao, P.D., Le Thi, H.A.: The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Ann. Oper. Res. 133, 23–46 (2005). https://doi.org/10.1007/s10479-004-5022-1
Toland, J.F.: Duality in nonconvex optimization. J. Math. Anal. Appl. 66(2), 399–415 (1978)
Yajima, Y., Fujie, T.A.: Polyhedral approach for nonconvex quadratic programming problems with box constraints. J. Global Optim. 13, 151–170 (1998). https://doi.org/10.1023/A:1008293029350
Acknowledgements
This work supported by the project “ P2019-3751” of National University of Mongolia.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Enkhbat, R., Gruzdeva, T.V., Enkhbayr, J. (2020). D.C. Constrained Optimization Approach for Solving Metal Recovery Processing Problem. In: Kononov, A., Khachay, M., Kalyagin, V., Pardalos, P. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2020. Lecture Notes in Computer Science(), vol 12095. Springer, Cham. https://doi.org/10.1007/978-3-030-49988-4_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-49988-4_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-49987-7
Online ISBN: 978-3-030-49988-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)