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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
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##### References:
 $x^1$ $x^2$ $\ldots$ $x^n$ $f^1$ $f^2$ $\ldots$ $f^l$ $x_{11}$ $x_{12}$ $\ldots$ $x_{1n}$ $f_{11}$ $f_{12}$ $\ldots$ $f_{1l}$ $x_{21}$ $x_{22}$ $\ldots$ $x_{2n}$ $f_{21}$ $f_{22}$ $\ldots$ $f_{2l}$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $x_{m1}$ $x_{m2}$ $\ldots$ $x_{m n}$ $f_{m1}$ $f_{m2}$ $\ldots$ $f_{ml}$
 $x^1$ $x^2$ $\ldots$ $x^n$ $f^1$ $f^2$ $\ldots$ $f^l$ $x_{11}$ $x_{12}$ $\ldots$ $x_{1n}$ $f_{11}$ $f_{12}$ $\ldots$ $f_{1l}$ $x_{21}$ $x_{22}$ $\ldots$ $x_{2n}$ $f_{21}$ $f_{22}$ $\ldots$ $f_{2l}$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $\ldots$ $x_{m1}$ $x_{m2}$ $\ldots$ $x_{m n}$ $f_{m1}$ $f_{m2}$ $\ldots$ $f_{ml}$
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