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Globally solving quadratic programs with convex objective and complementarity constraints via completely positive programming
April 2018, 14(2): 613-623. doi: 10.3934/jimo.2017063

## D.C. programming approach for solving an applied ore-processing problem

 1 Matrosov Institute for System Dynamics and Control Theory of SB RAS, Irkutsk, Russia 2 National University of Mongolia, Ulaanbaatar, Mongolia

* Corresponding author:renkhbat46@yahoo.com

Received  April 2016 Revised  December 2016 Published  June 2017

Fund Project: This work has been supported by the Russian Science Foundation, Project N15-11-2001

This paper was motivated by a practical optimization problem formulated at the Erdenet Mining Corporation (Mongolia). By solving an identification problem for a chosen design of experiment we developed a quadratic model that quite adequately represents the experimental data. The problem obtained turned out to be the indefinite quadratic program, which we solved by applying the global search theory for a d.c. programming developed by A.S. Strekalovsky [13]-[15]. According to this d.c. optimization theory, we performed a local search that takes into account the structure of the problem in question, and constructed procedures of escaping critical points provided by the local search. The algorithms proposed for d.c. programming were verified using a set of test problems as well as a copper content maximization problem arising at the mining factory.

Citation: R. Enkhbat, T. V. Gruzdeva, M. V. Barkova. D.C. programming approach for solving an applied ore-processing problem. Journal of Industrial & Management Optimization, 2018, 14 (2) : 613-623. doi: 10.3934/jimo.2017063
##### References:
 [1] R. Enkhbat and Ya. Bazarsad, General quadratic programming and its applications, in Optimization and optimal control (eds. A. Chinchuluun, P. M. Pardalos, R. Enkhbat and I. Tseveendorj), Springer-Verlag New York, (2010), 121-139. [2] R. Enkhbat and T. Ibaraki, Global optimization algorithms for general quadratic programming, J. Mong. Math. Soc., 5 (2001), 22-56. [3] T. V. Gruzdeva and A. S. Strekalovsky, Local search in problems with nonconvex constraints, Comput. Math. Math. Phys., 47 (2007), 381-396. doi: 10.1134/S0965542507030049. [4] R. Horst, P. Pardalos and N. V. Thoai, Introduction to Global Optimization, Introduction to Global Optimization, (1995). [5] V. Jeyakumar, G. M. Lee and N. T. H. Linh, Generalized Farkas lemma and gap-free duality for minimax dc optimization with polynomials and robust quadratic optimization, J. Global Optim., 64 (2016), 679-702. doi: 10.1007/s10898-015-0277-4. [6] V. Jeyakumar, A. M. Rubinov and Z. Y. Wu, Non-convex quadratic minimization problems with quadratic constraints: Global optimality conditions, Math. Program., 110 (2007), 521-541. doi: 10.1007/s10107-006-0012-5. [7] R. Horst and N. V. Thoai, D.C.programming: Overview, J. Optim. Theory Appl., 103 (1999), 1-43. doi: 10.1023/A:1021765131316. [8] R. H. Myers, Response Surface Methodology, Allyn and Bacon, Boston, MA, 1971. [9] J. Nocedal and S. J. Wright, Numerical Optimization, Springer-Verlag, New York, 1999. doi: 10.1007/b98874. [10] A. Rubinov, Abstract Convexity and Global Optimization, Springer US, Dordrecht, 2000. [11] A. M. Rubinov and Z. Y. Wu, Optimality conditions in global optimization and their applications, Math. Program., 120 (2009), 101-123. doi: 10.1007/s10107-007-0142-4. [12] A. S. Strekalovsky, On local search in d.c. optimization problems, Appl. Math. Comput., 255 (2015), 73-83. doi: 10.1016/j.amc.2014.08.092. [13] A. S. Strekalovsky, On solving optimization problems with hidden nonconvex structures, in Optimization in science and engineering (eds. T. M. Rassias, C. A. Floudas and S. Butenko), Springer, New York, (2014), 465-502. doi: 10.1007/978-1-4939-0808-0_23. [14] A. S. Strekalovsky, Elementy Nevypukloi Optimizatsii, (Russian) [Elements of nonconvex optimization], Nauka Publ., Novosibirsk, 2003. [15] A. S. Strekalovsky, On the minimization of the difference of convex functions on a feasible set, Comput. Math. Math. Phys., 43 (2003), 380-390. [16] A. S. Strekalovsky, A. A. Kuznetsova and T. V. Yakovleva, Numerical solution of nonconvex optimization problems, Numer. Anal. Appl., 4 (2001), 185-199. [17] A. S. Strekalovsky and T. V. Yakovleva, On a local and global search involved in nonconvex optimization problems, Autom. and Remote Control, 65 (2004), 375-387. doi: 10.1023/B:AURC.0000019368.45522.7a. [18] P. D. Tao and L. T. Hoai An, The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems, Ann. Oper. Res., 133 (2005), 23-46. doi: 10.1007/s10479-004-5022-1. [19] Z. Y. Wu, V. Jeyakumar and A. M. Rubinov, Sufficient conditions for global optimality of bivalent nonconvex quadratic programs with inequality constraints, J. Optim. Theory Appl., 133 (2007), 123-130. doi: 10.1007/s10957-007-9177-1. [20] Z. Y. Wu and A. M. Rubinov, Global optimality conditions for some classes of optimization problems, J. Optim. Theory Appl., 145 (2010), 164-185. doi: 10.1007/s10957-009-9616-2.

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##### References:
 [1] R. Enkhbat and Ya. Bazarsad, General quadratic programming and its applications, in Optimization and optimal control (eds. A. Chinchuluun, P. M. Pardalos, R. Enkhbat and I. Tseveendorj), Springer-Verlag New York, (2010), 121-139. [2] R. Enkhbat and T. Ibaraki, Global optimization algorithms for general quadratic programming, J. Mong. Math. Soc., 5 (2001), 22-56. [3] T. V. Gruzdeva and A. S. Strekalovsky, Local search in problems with nonconvex constraints, Comput. Math. Math. Phys., 47 (2007), 381-396. doi: 10.1134/S0965542507030049. [4] R. Horst, P. Pardalos and N. V. Thoai, Introduction to Global Optimization, Introduction to Global Optimization, (1995). [5] V. Jeyakumar, G. M. Lee and N. T. H. Linh, Generalized Farkas lemma and gap-free duality for minimax dc optimization with polynomials and robust quadratic optimization, J. Global Optim., 64 (2016), 679-702. doi: 10.1007/s10898-015-0277-4. [6] V. Jeyakumar, A. M. Rubinov and Z. Y. Wu, Non-convex quadratic minimization problems with quadratic constraints: Global optimality conditions, Math. Program., 110 (2007), 521-541. doi: 10.1007/s10107-006-0012-5. [7] R. Horst and N. V. Thoai, D.C.programming: Overview, J. Optim. Theory Appl., 103 (1999), 1-43. doi: 10.1023/A:1021765131316. [8] R. H. Myers, Response Surface Methodology, Allyn and Bacon, Boston, MA, 1971. [9] J. Nocedal and S. J. Wright, Numerical Optimization, Springer-Verlag, New York, 1999. doi: 10.1007/b98874. [10] A. Rubinov, Abstract Convexity and Global Optimization, Springer US, Dordrecht, 2000. [11] A. M. Rubinov and Z. Y. Wu, Optimality conditions in global optimization and their applications, Math. Program., 120 (2009), 101-123. doi: 10.1007/s10107-007-0142-4. [12] A. S. Strekalovsky, On local search in d.c. optimization problems, Appl. Math. Comput., 255 (2015), 73-83. doi: 10.1016/j.amc.2014.08.092. [13] A. S. Strekalovsky, On solving optimization problems with hidden nonconvex structures, in Optimization in science and engineering (eds. T. M. Rassias, C. A. Floudas and S. Butenko), Springer, New York, (2014), 465-502. doi: 10.1007/978-1-4939-0808-0_23. [14] A. S. Strekalovsky, Elementy Nevypukloi Optimizatsii, (Russian) [Elements of nonconvex optimization], Nauka Publ., Novosibirsk, 2003. [15] A. S. Strekalovsky, On the minimization of the difference of convex functions on a feasible set, Comput. Math. Math. Phys., 43 (2003), 380-390. [16] A. S. Strekalovsky, A. A. Kuznetsova and T. V. Yakovleva, Numerical solution of nonconvex optimization problems, Numer. Anal. Appl., 4 (2001), 185-199. [17] A. S. Strekalovsky and T. V. Yakovleva, On a local and global search involved in nonconvex optimization problems, Autom. and Remote Control, 65 (2004), 375-387. doi: 10.1023/B:AURC.0000019368.45522.7a. [18] P. D. Tao and L. T. Hoai An, The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems, Ann. Oper. Res., 133 (2005), 23-46. doi: 10.1007/s10479-004-5022-1. [19] Z. Y. Wu, V. Jeyakumar and A. M. Rubinov, Sufficient conditions for global optimality of bivalent nonconvex quadratic programs with inequality constraints, J. Optim. Theory Appl., 133 (2007), 123-130. doi: 10.1007/s10957-007-9177-1. [20] Z. Y. Wu and A. M. Rubinov, Global optimality conditions for some classes of optimization problems, J. Optim. Theory Appl., 145 (2010), 164-185. doi: 10.1007/s10957-009-9616-2.
 $x^1$ $x^2$ $\ldots$ $x^n$ $f^1$ $f^2$ $\ldots$ $f^l$ $x_{11}$ $x_{12}$ $\ldots$ $x_{1n}$ $f_{11}$ $f_{12}$ $\ldots$ $f_{1l}$ $x_{21}$ $x_{22}$ $\ldots$ $x_{2n}$ $f_{21}$ $f_{22}$ $\ldots$ $f_{2l}$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $x_{m1}$ $x_{m2}$ $\ldots$ $x_{m n}$ $f_{m1}$ $f_{m2}$ $\ldots$ $f_{ml}$
 $x^1$ $x^2$ $\ldots$ $x^n$ $f^1$ $f^2$ $\ldots$ $f^l$ $x_{11}$ $x_{12}$ $\ldots$ $x_{1n}$ $f_{11}$ $f_{12}$ $\ldots$ $f_{1l}$ $x_{21}$ $x_{22}$ $\ldots$ $x_{2n}$ $f_{21}$ $f_{22}$ $\ldots$ $f_{2l}$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $x_{m1}$ $x_{m2}$ $\ldots$ $x_{m n}$ $f_{m1}$ $f_{m2}$ $\ldots$ $f_{ml}$
Local search method for Problem $(P_1)$
 $#$ $x^0$ $f_1(x^0)$ $f_1(z)$ $PL$ $Time$ $1$ $(0.408, 1.000, 0.572, 1.000, 0.628, 1.000, 0.167)$ $0.91617$ $0.93224$ $6$ $0.062$ $2$ $(0.408, 1.000, 1.000, 1.000, 1.000, 1.000, 1.000 )$ $1.10581$ $1.28877$ $6$ $0.076$ $3$ $(1.000, 0.000, 1.000, 1.000, 1.000, 1.000, 1.000)$ $1.20652$ $1.35330$ $5$ $0.047$ $4$ $(0.987, 0.920, 0.852, 0.914, 0.893, 0.796, 0.186)$ $0.87444$ $1.36455$ $7$ $0.015$ $5$ $(0.658, 0.699, 0.970, 0.783, 0.629, 0.858, 0.847)$ $0.83431$ ${\bf{1.36510}}$ $\!10\!$ $0.010$
 $#$ $x^0$ $f_1(x^0)$ $f_1(z)$ $PL$ $Time$ $1$ $(0.408, 1.000, 0.572, 1.000, 0.628, 1.000, 0.167)$ $0.91617$ $0.93224$ $6$ $0.062$ $2$ $(0.408, 1.000, 1.000, 1.000, 1.000, 1.000, 1.000 )$ $1.10581$ $1.28877$ $6$ $0.076$ $3$ $(1.000, 0.000, 1.000, 1.000, 1.000, 1.000, 1.000)$ $1.20652$ $1.35330$ $5$ $0.047$ $4$ $(0.987, 0.920, 0.852, 0.914, 0.893, 0.796, 0.186)$ $0.87444$ $1.36455$ $7$ $0.015$ $5$ $(0.658, 0.699, 0.970, 0.783, 0.629, 0.858, 0.847)$ $0.83431$ ${\bf{1.36510}}$ $\!10\!$ $0.010$
Local search method for Problem $(P_2)$
 $#$ $x^0$ $f_2(x^0)$ $f_2(z)$ $PL$ $Time$ $1$ $(1.000, 1.000, 1.000, 1.000, 1.000, 1.000, 1.000)$ $0.87224$ ${\bf{1.10128}}$ $9$ $0.090$ $2$ $(0.408, 1.000, 0.572, 1.000, 0.628, 1.000, 0.167)$ $1.02257$ $1.04541$ $5$ $0.047$ $3$ $(0.408, 1.000, 1.000, 1.000, 1.000, 1.000, 1.000)$ $1.08494$ ${\bf{1.10126}}$ $8$ $0.078$ $4$ $(0.408, 0.000, 0.572, 0.724, 0.628, 1.000, 0.167)$ $0.93835$ $1.10021$ $\!16\!$ $0.031$ $5$ $(1.000, 1.000, 1.000, 1.000, 1.000, 1.000, 0.167)$ 0.99559 1.09847 8 0.012
 $#$ $x^0$ $f_2(x^0)$ $f_2(z)$ $PL$ $Time$ $1$ $(1.000, 1.000, 1.000, 1.000, 1.000, 1.000, 1.000)$ $0.87224$ ${\bf{1.10128}}$ $9$ $0.090$ $2$ $(0.408, 1.000, 0.572, 1.000, 0.628, 1.000, 0.167)$ $1.02257$ $1.04541$ $5$ $0.047$ $3$ $(0.408, 1.000, 1.000, 1.000, 1.000, 1.000, 1.000)$ $1.08494$ ${\bf{1.10126}}$ $8$ $0.078$ $4$ $(0.408, 0.000, 0.572, 0.724, 0.628, 1.000, 0.167)$ $0.93835$ $1.10021$ $\!16\!$ $0.031$ $5$ $(1.000, 1.000, 1.000, 1.000, 1.000, 1.000, 0.167)$ 0.99559 1.09847 8 0.012
Global search method for Problem $(P_1)$
 $#$ $f_1(x^0)$ $f_1^*$ $it$ $loc$ $PL$ $Time$ 1 0.91617 1.36518 8 161 346 0.260 2 1.10581 1.36518 9 157 338 0.250 3 1.20652 1.36518 9 165 353 0.262 4 0.87444 1.36518 5 147 319 0.234 5 0.83431 1.36518 1 136 294 0.218
 $#$ $f_1(x^0)$ $f_1^*$ $it$ $loc$ $PL$ $Time$ 1 0.91617 1.36518 8 161 346 0.260 2 1.10581 1.36518 9 157 338 0.250 3 1.20652 1.36518 9 165 353 0.262 4 0.87444 1.36518 5 147 319 0.234 5 0.83431 1.36518 1 136 294 0.218
Global search method for Problem $(P_2)$
 $#$ $f_2(x^0)$ $f_2^*$ $it$ $loc$ $PL$ $Time$ 1 0.87224 1.10128 1 74 145 0.124 2 1.02257 1.10128 8 91 199 0.171 3 1.08494 1.10128 1 74 155 0.124 4 0.93835 1.10128 6 85 188 0.141 5 0.99559 1.10128 8 91 199 0.156
 $#$ $f_2(x^0)$ $f_2^*$ $it$ $loc$ $PL$ $Time$ 1 0.87224 1.10128 1 74 145 0.124 2 1.02257 1.10128 8 91 199 0.171 3 1.08494 1.10128 1 74 155 0.124 4 0.93835 1.10128 6 85 188 0.141 5 0.99559 1.10128 8 91 199 0.156

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