• Previous Article
    Globally solving quadratic programs with convex objective and complementarity constraints via completely positive programming
  • JIMO Home
  • This Issue
  • Next Article
    An optimized direction statistics for detecting and removing random-valued impulse noise
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. Google Scholar

[2]

R. Enkhbat and T. Ibaraki, Global optimization algorithms for general quadratic programming, J. Mong. Math. Soc., 5 (2001), 22-56.   Google Scholar

[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.  Google Scholar

[4]

R. HorstP. Pardalos and N. V. Thoai, Introduction to Global Optimization, Introduction to Global Optimization, (1995).   Google Scholar

[5]

V. JeyakumarG. 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.  Google Scholar

[6]

V. JeyakumarA. 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.  Google Scholar

[7]

R. Horst and N. V. Thoai, D.C.programming: Overview, J. Optim. Theory Appl., 103 (1999), 1-43.  doi: 10.1023/A:1021765131316.  Google Scholar

[8]

R. H. Myers, Response Surface Methodology, Allyn and Bacon, Boston, MA, 1971. Google Scholar

[9]

J. Nocedal and S. J. Wright, Numerical Optimization, Springer-Verlag, New York, 1999. doi: 10.1007/b98874.  Google Scholar

[10]

A. Rubinov, Abstract Convexity and Global Optimization, Springer US, Dordrecht, 2000. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. S. Strekalovsky, Elementy Nevypukloi Optimizatsii, (Russian) [Elements of nonconvex optimization], Nauka Publ., Novosibirsk, 2003. Google Scholar

[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.   Google Scholar

[16]

A. S. StrekalovskyA. A. Kuznetsova and T. V. Yakovleva, Numerical solution of nonconvex optimization problems, Numer. Anal. Appl., 4 (2001), 185-199.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

Z. Y. WuV. 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.  Google Scholar

[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.  Google Scholar

show all references

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. Google Scholar

[2]

R. Enkhbat and T. Ibaraki, Global optimization algorithms for general quadratic programming, J. Mong. Math. Soc., 5 (2001), 22-56.   Google Scholar

[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.  Google Scholar

[4]

R. HorstP. Pardalos and N. V. Thoai, Introduction to Global Optimization, Introduction to Global Optimization, (1995).   Google Scholar

[5]

V. JeyakumarG. 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.  Google Scholar

[6]

V. JeyakumarA. 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.  Google Scholar

[7]

R. Horst and N. V. Thoai, D.C.programming: Overview, J. Optim. Theory Appl., 103 (1999), 1-43.  doi: 10.1023/A:1021765131316.  Google Scholar

[8]

R. H. Myers, Response Surface Methodology, Allyn and Bacon, Boston, MA, 1971. Google Scholar

[9]

J. Nocedal and S. J. Wright, Numerical Optimization, Springer-Verlag, New York, 1999. doi: 10.1007/b98874.  Google Scholar

[10]

A. Rubinov, Abstract Convexity and Global Optimization, Springer US, Dordrecht, 2000. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. S. Strekalovsky, Elementy Nevypukloi Optimizatsii, (Russian) [Elements of nonconvex optimization], Nauka Publ., Novosibirsk, 2003. Google Scholar

[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.   Google Scholar

[16]

A. S. StrekalovskyA. A. Kuznetsova and T. V. Yakovleva, Numerical solution of nonconvex optimization problems, Numer. Anal. Appl., 4 (2001), 185-199.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

Z. Y. WuV. 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.  Google Scholar

[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.  Google Scholar

Table 1.   
$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}$
Table 1.  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$
Table 2.  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\!$$0.031$
$5$ $(1.000, 1.000, 1.000, 1.000, 1.000, 1.000, 0.167) $0.995591.0984780.012
$#$$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) $0.995591.0984780.012
Table 3.  Global search method for Problem $(P_1)$
$#$$f_1(x^0)$$f_1^*$$it$$loc$$PL$$Time$
10.916171.3651881613460.260
21.105811.3651891573380.250
31.206521.3651891653530.262
40.874441.3651851473190.234
50.834311.3651811362940.218
$#$$f_1(x^0)$$f_1^*$$it$$loc$$PL$$Time$
10.916171.3651881613460.260
21.105811.3651891573380.250
31.206521.3651891653530.262
40.874441.3651851473190.234
50.834311.3651811362940.218
Table 4.  Global search method for Problem $(P_2)$
$#$$f_2(x^0)$$f_2^*$$it$$loc$$PL$$Time$
10.872241.101281741450.124
21.022571.101288911990.171
31.084941.101281741550.124
40.938351.101286851880.141
50.995591.101288911990.156
$#$$f_2(x^0)$$f_2^*$$it$$loc$$PL$$Time$
10.872241.101281741450.124
21.022571.101288911990.171
31.084941.101281741550.124
40.938351.101286851880.141
50.995591.101288911990.156
[1]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212

[2]

Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393

[3]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023

[4]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[5]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881

[6]

Valeria Chiado Piat, Sergey S. Nazarov, Andrey Piatnitski. Steklov problems in perforated domains with a coefficient of indefinite sign. Networks & Heterogeneous Media, 2012, 7 (1) : 151-178. doi: 10.3934/nhm.2012.7.151

[7]

Jon Aaronson, Dalia Terhesiu. Local limit theorems for suspended semiflows. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6575-6609. doi: 10.3934/dcds.2020294

[8]

Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190

[9]

Jean-François Biasse. Improvements in the computation of ideal class groups of imaginary quadratic number fields. Advances in Mathematics of Communications, 2010, 4 (2) : 141-154. doi: 10.3934/amc.2010.4.141

[10]

Baba Issa Camara, Houda Mokrani, Evans K. Afenya. Mathematical modeling of glioma therapy using oncolytic viruses. Mathematical Biosciences & Engineering, 2013, 10 (3) : 565-578. doi: 10.3934/mbe.2013.10.565

[11]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094

[12]

Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034

[13]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072

[14]

Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009

[15]

Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933

[16]

Ronald E. Mickens. Positivity preserving discrete model for the coupled ODE's modeling glycolysis. Conference Publications, 2003, 2003 (Special) : 623-629. doi: 10.3934/proc.2003.2003.623

[17]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397

[18]

Brandy Rapatski, James Yorke. Modeling HIV outbreaks: The male to female prevalence ratio in the core population. Mathematical Biosciences & Engineering, 2009, 6 (1) : 135-143. doi: 10.3934/mbe.2009.6.135

[19]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617

[20]

Christina Surulescu, Nicolae Surulescu. Modeling and simulation of some cell dispersion problems by a nonparametric method. Mathematical Biosciences & Engineering, 2011, 8 (2) : 263-277. doi: 10.3934/mbe.2011.8.263

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (111)
  • HTML views (791)
  • Cited by (1)

Other articles
by authors

[Back to Top]