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September  2020, 16(5): 2351-2367. doi: 10.3934/jimo.2019057

A new concave reformulation and its application in solving DC programming globally under uncertain environment

Yanjun Wang , and  Kaiji Shen

School of Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, China

* Corresponding author: Yanjun Wang

Received  July 2018 Revised  January 2019 Published  May 2019

Fund Project: We gratefully acknowledge the valuable cooperation of Prof. R. Tyrrell Rockafellar(University of Washington). This research was supported by NSFC (11271243)

In this paper, a new concave reformulation is proposed on a convex hull of some given points. Based on its properties, we attempt to solve DC Programming problems globally under uncertain environment by using Robust optimization method and CVaR method. A global optimization algorithm is developed for the Robust counterpart and CVaR model with two kinds of special convex hulls: simplex set and box set. The global solution is obtained by solving a sequence of convex relaxation programming on the original constraint sets or divided subsets with branch and bound method. Finally, numerical experiments are given for DC programs under uncertain environment with two kinds of constraints: simplex and box sets. Simulation results show the feasibility and efficiency of the proposed global optimization algorithm.

Keywords: Concave reformulation, DC programming, global optimization, convex relaxation, branch and bound.
Mathematics Subject Classification: Primary: 65K05; Secondary: 90C90.
Citation: Yanjun Wang, Kaiji Shen. A new concave reformulation and its application in solving DC programming globally under uncertain environment. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2351-2367. doi: 10.3934/jimo.2019057
References:
[1]

A. Ben-Tal and A. Nemirovski, Robust convex optimization, Mathematics of Operations Research, 23 (1998), 769-805.  doi: 10.1287/moor.23.4.769.  Google Scholar

[2]

A. Ben-Tal and A. Nemirovski, Robust solutions of uncertain linear programs, Operations research letters, 25 (1999), 1-13.  doi: 10.1016/S0167-6377(99)00016-4.  Google Scholar

[3]

A. Ben-Tal and A. Nemirovski, Robust solutions of linear programming problems contaminated with uncertain data, Mathematical Programming, 88 (2000), 411-424.  doi: 10.1007/PL00011380.  Google Scholar

[4]

T. P. Dinh, The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems, Annals of Operations Research, 113 (2005), 23-46.  doi: 10.1007/s10479-004-5022-1.  Google Scholar

[5]

T. P. Dinh and A. Le Thi Hoai, A DC optimization algorithm for solving the trust-region subproblem, SIAM J. Optim., 8 (1998), 476-505.  doi: 10.1137/S1052623494274313.  Google Scholar

[6]

A. Edward, H. F. Xu and D. L. Zhang, Confidence levels for cvar risk measures and minimax limits, Business Analytics, 2014. Google Scholar

[7]

R. Horst and H. Tuy, Global Optimization: Deterministic Approaches, Second edition. Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02947-3.  Google Scholar

[8]

C. D. Maranas and C. A. Floudas, Global optimization in generalized geometric programming, Computers & Chemical Engineering, 21 (1997), 351-369.   Google Scholar

[9]

G. C. Pflug, Some remarks on the value-at-risk and the conditional value-at-risk, Probabilistic constrained optimization, 8 (2000), 272-281.  doi: 10.1007/978-1-4757-3150-7_15.  Google Scholar

[10]

H. Reiner and T. Nguyen V, Robust solutions of linear programming problems contaminated with uncertain data, Journal of Optimization Theory and Applications, 103 (1999), 1-43.   Google Scholar

[11] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N.J., 1970.   Google Scholar
[12]

R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 21-41.  doi: 10.21314/JOR.2000.038.  Google Scholar

[13]

R. T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distributions, Journal of Banking & Finance, 26 (2002), 1443-1471.   Google Scholar

[14]

R. T. Rockafellar, Coherent approaches to risk in optimization under uncertainty, OR Tools and Applications: Glimpses of Future Technologies, 8 (2007), 38-61.  doi: 10.1287/educ.1073.0032.  Google Scholar

[15]

P. P. Shen and C. F. Wang, Global optimization for sum of linear ratios problem with coefficients, Applied Mathematics and Computation, 176 (2006), 219-229.  doi: 10.1016/j.amc.2005.09.047.  Google Scholar

[16]

Y. J. Wang and Y. Lan, Global optimization for special reverse convex programming, Computers & Mathematics with Applications, 55 (2008), 1154-1163.  doi: 10.1016/j.camwa.2007.04.046.  Google Scholar

show all references

References:
[1]

A. Ben-Tal and A. Nemirovski, Robust convex optimization, Mathematics of Operations Research, 23 (1998), 769-805.  doi: 10.1287/moor.23.4.769.  Google Scholar

[2]

A. Ben-Tal and A. Nemirovski, Robust solutions of uncertain linear programs, Operations research letters, 25 (1999), 1-13.  doi: 10.1016/S0167-6377(99)00016-4.  Google Scholar

[3]

A. Ben-Tal and A. Nemirovski, Robust solutions of linear programming problems contaminated with uncertain data, Mathematical Programming, 88 (2000), 411-424.  doi: 10.1007/PL00011380.  Google Scholar

[4]

T. P. Dinh, The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems, Annals of Operations Research, 113 (2005), 23-46.  doi: 10.1007/s10479-004-5022-1.  Google Scholar

[5]

T. P. Dinh and A. Le Thi Hoai, A DC optimization algorithm for solving the trust-region subproblem, SIAM J. Optim., 8 (1998), 476-505.  doi: 10.1137/S1052623494274313.  Google Scholar

[6]

A. Edward, H. F. Xu and D. L. Zhang, Confidence levels for cvar risk measures and minimax limits, Business Analytics, 2014. Google Scholar

[7]

R. Horst and H. Tuy, Global Optimization: Deterministic Approaches, Second edition. Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02947-3.  Google Scholar

[8]

C. D. Maranas and C. A. Floudas, Global optimization in generalized geometric programming, Computers & Chemical Engineering, 21 (1997), 351-369.   Google Scholar

[9]

G. C. Pflug, Some remarks on the value-at-risk and the conditional value-at-risk, Probabilistic constrained optimization, 8 (2000), 272-281.  doi: 10.1007/978-1-4757-3150-7_15.  Google Scholar

[10]

H. Reiner and T. Nguyen V, Robust solutions of linear programming problems contaminated with uncertain data, Journal of Optimization Theory and Applications, 103 (1999), 1-43.   Google Scholar

[11] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N.J., 1970.   Google Scholar
[12]

R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 21-41.  doi: 10.21314/JOR.2000.038.  Google Scholar

[13]

R. T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distributions, Journal of Banking & Finance, 26 (2002), 1443-1471.   Google Scholar

[14]

R. T. Rockafellar, Coherent approaches to risk in optimization under uncertainty, OR Tools and Applications: Glimpses of Future Technologies, 8 (2007), 38-61.  doi: 10.1287/educ.1073.0032.  Google Scholar

[15]

P. P. Shen and C. F. Wang, Global optimization for sum of linear ratios problem with coefficients, Applied Mathematics and Computation, 176 (2006), 219-229.  doi: 10.1016/j.amc.2005.09.047.  Google Scholar

[16]

Y. J. Wang and Y. Lan, Global optimization for special reverse convex programming, Computers & Mathematics with Applications, 55 (2008), 1154-1163.  doi: 10.1016/j.camwa.2007.04.046.  Google Scholar

Figure 1.  h(x,ω0) and l(x,ω0) in box case
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Figure 2.  h(x,ω0) and l(x,ω0) in simplex case
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Figure 3.  Optimal value comparison of Robust Model and CVaR model
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Table 1.   
α CPU(s) Step Nodes Opt Solution Opt Value Opt*
0.70 356.11 87 69 (0.0000, 1.1250, 1.8750)T -2.2690 -2.2689
0.75 956.11 163 101 (0.0000, 1.0547, 1.5703)T -2.1214 -2.1214
0.80 847.73 163 106 (0.0000, 0.7969, 1.1191)T -1.9628 -1.9628
0.85 516.92 132 106 (0.0000, 0.6211, 0.8789)T -1.7508 -1.7508
0.90 518.31 122 107 (0.0000, 0.4569, 0.6797)T -1.5661 -1.5663
0.95 321.43 78 68 (0.0000, 0.3580, 0.5326)T -1.4453 -1.4452
0.97 143.17 31 27 (0.0000, 0.3387, 0.5038)T -1.4228 -1.4228
0.98 160.39 30 24 (0.0104, 0.3437, 0.5156)T -1.4222 -1.4222
0.99 167.81 31 26 (0.0023, 0.3281, 0.5000)T -1.4221 -1.4221
Robust 218.18 52 51 (0.0080, 0.3515, 0.5002)T -1.4221 -1.4221
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α CPU(s) Step Nodes Opt Solution Opt Value Opt*
0.70 356.11 87 69 (0.0000, 1.1250, 1.8750)T -2.2690 -2.2689
0.75 956.11 163 101 (0.0000, 1.0547, 1.5703)T -2.1214 -2.1214
0.80 847.73 163 106 (0.0000, 0.7969, 1.1191)T -1.9628 -1.9628
0.85 516.92 132 106 (0.0000, 0.6211, 0.8789)T -1.7508 -1.7508
0.90 518.31 122 107 (0.0000, 0.4569, 0.6797)T -1.5661 -1.5663
0.95 321.43 78 68 (0.0000, 0.3580, 0.5326)T -1.4453 -1.4452
0.97 143.17 31 27 (0.0000, 0.3387, 0.5038)T -1.4228 -1.4228
0.98 160.39 30 24 (0.0104, 0.3437, 0.5156)T -1.4222 -1.4222
0.99 167.81 31 26 (0.0023, 0.3281, 0.5000)T -1.4221 -1.4221
Robust 218.18 52 51 (0.0080, 0.3515, 0.5002)T -1.4221 -1.4221
Table 2.   
α CPU(s) Step Nodes Opt Solution Opt Value Opt*
0.70 264.90 24 22 (0.0000, 0.6719, 0.9844)T -2.2410 -2.2410
0.75 105.19 25 29 (0.0000, 0.6641, 0.9844)T -2.1214 -2.1215
0.80 217.65 24 19 (0.0000, 0.7187, 0.9844)T -1.9520 -1.9520
0.85 516.92 55 40 (0.0000, 0.6250, 0.8750)T -1.7506 -1.7505
0.90 350.26 39 34 (0.0000, 0.4531, 0.6741)T -1.5661 -1.5661
0.95 208.85 38 32 (0.0000, 0.3580, 0.5326)T -1.4452 -1.4452
0.97 143.17 31 27 (0.0000, 0.3437, 0.5156)T -1.4228 -1.4228
0.98 167.81 30 26 (0.0104, 0.3437, 0.5156)T -1.4222 -1.4222
0.99 160.39 31 24 (0.0023, 0.3281, 0.5000)T -1.4221 -1.4221
Robust 216.98 35 27 (0.0080, 0.3515, 0.5002)T -1.4221 -1.4221
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α CPU(s) Step Nodes Opt Solution Opt Value Opt*
0.70 264.90 24 22 (0.0000, 0.6719, 0.9844)T -2.2410 -2.2410
0.75 105.19 25 29 (0.0000, 0.6641, 0.9844)T -2.1214 -2.1215
0.80 217.65 24 19 (0.0000, 0.7187, 0.9844)T -1.9520 -1.9520
0.85 516.92 55 40 (0.0000, 0.6250, 0.8750)T -1.7506 -1.7505
0.90 350.26 39 34 (0.0000, 0.4531, 0.6741)T -1.5661 -1.5661
0.95 208.85 38 32 (0.0000, 0.3580, 0.5326)T -1.4452 -1.4452
0.97 143.17 31 27 (0.0000, 0.3437, 0.5156)T -1.4228 -1.4228
0.98 167.81 30 26 (0.0104, 0.3437, 0.5156)T -1.4222 -1.4222
0.99 160.39 31 24 (0.0023, 0.3281, 0.5000)T -1.4221 -1.4221
Robust 216.98 35 27 (0.0080, 0.3515, 0.5002)T -1.4221 -1.4221
Table 3.  Simplex feasible
CPU Time(s) Nodes SEED
RDC 91 11 1
Floudas 227 42 1
RDC 100 15 2
Floudas 204 40 2
RDC 120 25 3
Floudas 280 75 3
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CPU Time(s) Nodes SEED
RDC 91 11 1
Floudas 227 42 1
RDC 100 15 2
Floudas 204 40 2
RDC 120 25 3
Floudas 280 75 3
Table 4.  Box feasible
CPU Time(s) Nodes SEED
RDC 88 8 1
Floudas 220 40 1
RDC 105 12 2
Floudas 207 36 2
RDC 117 20 3
Floudas 281 68 3
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CPU Time(s) Nodes SEED
RDC 88 8 1
Floudas 220 40 1
RDC 105 12 2
Floudas 207 36 2
RDC 117 20 3
Floudas 281 68 3
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