
-
Previous Article
Analysis of optimal boundary control for a three-dimensional reaction-diffusion system
- NACO Home
- This Issue
-
Next Article
On the pinning controllability of complex networks using perturbation theory of extreme singular values. application to synchronisation in power grids
A simplex grey wolf optimizer for solving integer programming and minimax problems
1. | Department of Mathematics and Statistics, Faculty of Science, Thompson Rivers University, Kamloops, BC, Canada V2C 0C8 |
2. | Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Moharam Bey 21511, Alexandria, Egypt |
3. | Department of Computer Science, Faculty of Computers & Informatics, Suez Canal University, Ismailia, Egypt21511, Alexandria, Egypt |
4. | Postdoctoral fellow, Department of Mathematics and Statistics, Faculty of Science, Thompson Rivers University, Kamloops, BC, Canada V2C 0C8 |
In this paper, we propose a new hybrid grey wolf optimizer (GWO) algorithm with simplex Nelder-Mead method in order to solve integer programming and minimax problems. We call the proposed algorithm a Simplex Grey Wolf Optimizer (SGWO) algorithm. In the the proposed SGWO algorithm, we combine the GWO algorithm with the Nelder-Mead method in order to refine the best obtained solution from the standard GWO algorithm. We test it on 7 integer programming problems and 10 minimax problems in order to investigate the general performance of the proposed SGWO algorithm. Also, we compare SGWO with 10 algorithms for solving integer programming problems and 9 algorithms for solving minimax problems. The experiments results show the efficiency of the proposed algorithm and its ability to solve integer and minimax optimization problems in reasonable time.
References:
[1] |
J. S. Arora, Introduction to Optimum Design, McGraw–Hill, New York, 1989. |
[2] |
N. Bacanin and M. Tuba,
Artificial Bee Colony (ABC) algorithm for constrained optimization improved with genetic operators, Studies in Informatics and Control, 21 (2012), 137-146.
|
[3] |
N. Bacanin, I. Brajevic and M. Tuba, Firefly Algorithm Applied to Integer Programming Problems, Recent Advances in Mathematics, 2013. |
[4] |
J. W. Bandler and C. Charalambous,
Nonlinear programming using minimax techniques, Journal of Optimization Theory and Applications, 13 (1974), 607-619.
doi: 10.1007/BF00933620. |
[5] |
B. Borchers and J. E. Mitchell, Using an interior point method in a branch and bound algorithm for integer programming, Technical Report, Rensselaer Polytechnic Institute, July 1992.
doi: 10.1016/0305-0548(94)90024-8. |
[6] |
S. A. Chu, P. -W. Tsai and J. -S. Pan. Cat swarm optimization, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 4099 (2006), 854–858. |
[7] |
M. Dorigo, Optimization, Learning and Natural Algorithms, Ph. D. Thesis, Politecnico di Milano, Italy, 1992. |
[8] |
D. Z. Du and P. M. Pardalose, Minimax and Applications, Kluwer, 1995.
doi: 10.1007/978-1-4613-3557-3. |
[9] |
R. Fletcher, Practical Method of Optimization, Vol. 1 & 2, John Wiley and Sons, 1980. |
[10] |
A. Glankwahmdee, J. S. Liebman and G. L. Hogg,
Unconstrained discrete nonlinear programming, Engineering Optimization, 4 (1979), 95-107.
|
[11] |
P. E. Gill, W. Murray and M. H. Wright, Practical Optimization, Academic Press, London, 1981.
![]() ![]() |
[12] |
J. H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, MI, 1975.
![]() ![]() |
[13] |
A. C. P. Isabel, E. Santo and E. Fernandes,
Heuristics pattern search for bound constrained minimax problems, Computational Science and Its Applications, ICCSA, 6784 (2011), 174-184.
doi: 10.1007/978-3-642-21931-3_15. |
[14] |
R. Jovanovic and M. Tuba,
An ant colony optimization algorithm with improved pheromone correction strategy for the minimum weight vertex cover problem, Applied Soft Computing, 11 (2011), 5360-5366.
|
[15] |
R. Jovanovic and M. Tuba,
Ant colony optimization algorithm with pheromone correction strategy for minimum connected dominating set problem, Computer Science and Information Systems (ComSIS), 10 (2013), 133-149.
doi: 10.2298/CSIS110927038J. |
[16] |
D. Karaboga and B. Basturk,
A powerful and efficient algorithm for numerical function optimization: artificial bee colony (abc) algorithm, Journal of Global Optimization, 39 (2007), 459-471.
doi: 10.1007/s10898-007-9149-x. |
[17] |
J. Kennedy and R. C. Eberhart,
Particle Swarm Optimization, Proceedings of the IEEE International Conference on Neural Networks, 4 (1995), 1942-1948.
|
[18] |
E. C. Laskari, K. E. Parsopoulos and M. N. Vrahatis, Particle swarm optimization for integer programming, Proceedings of the IEEE 2002 Congress on Evolutionary Computation, Honolulu (HI), (2002), 1582–1587. |
[19] |
E. L. Lawler and D. W. Wood,
Branch and bound methods: A survey, Operations Research, 14 (1966), 699-719.
|
[20] |
X. L. Li, Z. J. Shao and J. X. Qian,
An optimizing method based on autonomous animals: Fish-swarm algorithm, System Engineering Theory and Practice, 22 (2003), 32-38.
|
[21] |
G. Liuzzi, S. Lucidi and M. Sciandrone,
A derivative-free algorithm for linearly constrained finite minimax problems, SIAM Journal on Optimization, 16 (2006), 1054-1075.
doi: 10.1137/040615821. |
[22] |
L. Lukan and J. Vlcek, Test problems for nonsmooth unconstrained and linearly constrained optimization, Technical report 798, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic, 2000. |
[23] |
V. M. Manquinho, J. P. Marques Silva, A. L. Oliveira and K. A. Sakallah, Branch and bound algorithms for highly constrained integer programs, Technical Report, Cadence European Laboratories, Portugal, 1997. |
[24] |
S. Mirjalili, S. M. Mirjalili and A. Lewis,
Grey wolf optimizer, Advances in Engineering Software, 69 (2014), 46-61.
|
[25] |
J. A. Nelder and R. Mead,
A simplex method for function minimization, Computer Journal, 7 (1965), 308-313.
doi: 10.1093/comjnl/7.4.308. |
[26] |
G. L. Nemhauser, A. H. G. Rinnooy Kan and M. J. Todd, Handbooks in OR & MS, volume 1. Elsevier, 1989. |
[27] |
K. E. Parsopoulos and M. N. Vrahatis, Unified particle swarm optimization for tackling operations research problems, in Proceeding of IEEE 2005 swarm Intelligence Symposium, Pasadena, USA, (2005), 53-59. |
[28] |
M. K. Passino,
Biomimicry of bacterial foraging for distributed optimization and control, Control Systems, IEEE, 22 (2002), 52-67.
|
[29] |
Y. G. Petalas, K. E. Parsopoulos and M. N. Vrahatis,
Memetic particle swarm optimization, Ann oper Res, 156 (2007), 99-127.
doi: 10.1007/s10479-007-0224-y. |
[30] |
E. Polak, J. O. Royset and R. S. Womersley,
Algorithms with adaptive smoothing for finite minimax problems, Journal of Optimization Theory and Applications, 119 (2003), 459-484.
doi: 10.1023/B:JOTA.0000006685.60019.3e. |
[31] |
S. S. Rao, Engineering Optimization-Theory and Practice, Wiley: New Delhi, 1994. |
[32] |
G. Rudolph, An evolutionary algorithm for integer programming, in Parallel Problem Solving from Nature(eds. Y. Davidor, H-P. Schwefel, and R. Mnner), 3 (1994), 139-148. |
[33] |
E. Sandgen,
Nonlinear integer and discrete programming in mechanical design optimization, Journal of Mechanical Design (ASME), 112 (1990), 223-229.
|
[34] |
H. P. Schwefel, Evolution and Optimum Seeking, New York: Wiley, 1995. |
[35] |
R. Storn and K. Price,
Differential evolution simple and efficient heuristic for global optimization over continuous spaces, J. Glob. Optim., 11 (1997), 341-359.
doi: 10.1023/A:1008202821328. |
[36] |
R. Tang, S. Fong, X. S. Yang and S. Deb, Wolf search algorithm with ephemeral memory, In Digital Information Management (ICDIM), 2012 Seventh International Conference on Digital Information Management, (2012), 165–172. |
[37] |
D. Teodorovic and M. DellOrco. Bee colony optimization cooperative learning approach to complex transportation problems, In Advanced OR and AI Methods in Transportation, Proceedings of 16th MiniEURO Conference and 10th Meeting of EWGT (13-16 September 2005). Poznan: Publishing House of the Polish Operational and System Research, (2005), 51–60. |
[38] |
M. Tuba, N. Bacanin and N. Stanarevic,
Adjusted artificial bee colony (ABC) algorithm for engineering problems, WSEAS Transaction on Computers, 1 (2012), 111-120.
|
[39] |
M. Tuba, M. Subotic and N. Stanarevic,
Performance of a modified cuckoo search algorithm for unconstrained optimization problems, WSEAS Transactions on Systems, 11 (2012), 62-74.
|
[40] |
B. Wilson, A Simplicial Algorithm for Concave Programming, PhD thesis, Harvard University, 1963. |
[41] |
S. Xu,
Smoothing method for minimax problems, Computational Optimization and Applications, 20 (2001), 267-279.
doi: 10.1023/A:1011211101714. |
[42] |
X. S. Yang,
A new metaheuristic bat-inspired algorithm, Nature Inspired Cooperative Strategies for Optimization (NICSO 2010), (2010), 65-74.
|
[43] |
X. S. Yang,
Firefly algorithm, stochastic test functions and design optimisation, International Journal of Bio-Inspired Computation, 2 (2010), 78-84.
|
[44] |
X. S. Yang and S. Deb, Cuckoo search via levy fights, in Nature & Biologically Inspired Computing, NaBIC 2009, World Congress on, IEEE, (2009), 210–214. |
[45] |
Z. Shen, A. Neumaier and M. C. Eiermann,
Solving minimax problems by interval methods, BIT, 30 (1990), 742-751.
doi: 10.1007/BF01933221. |
show all references
References:
[1] |
J. S. Arora, Introduction to Optimum Design, McGraw–Hill, New York, 1989. |
[2] |
N. Bacanin and M. Tuba,
Artificial Bee Colony (ABC) algorithm for constrained optimization improved with genetic operators, Studies in Informatics and Control, 21 (2012), 137-146.
|
[3] |
N. Bacanin, I. Brajevic and M. Tuba, Firefly Algorithm Applied to Integer Programming Problems, Recent Advances in Mathematics, 2013. |
[4] |
J. W. Bandler and C. Charalambous,
Nonlinear programming using minimax techniques, Journal of Optimization Theory and Applications, 13 (1974), 607-619.
doi: 10.1007/BF00933620. |
[5] |
B. Borchers and J. E. Mitchell, Using an interior point method in a branch and bound algorithm for integer programming, Technical Report, Rensselaer Polytechnic Institute, July 1992.
doi: 10.1016/0305-0548(94)90024-8. |
[6] |
S. A. Chu, P. -W. Tsai and J. -S. Pan. Cat swarm optimization, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 4099 (2006), 854–858. |
[7] |
M. Dorigo, Optimization, Learning and Natural Algorithms, Ph. D. Thesis, Politecnico di Milano, Italy, 1992. |
[8] |
D. Z. Du and P. M. Pardalose, Minimax and Applications, Kluwer, 1995.
doi: 10.1007/978-1-4613-3557-3. |
[9] |
R. Fletcher, Practical Method of Optimization, Vol. 1 & 2, John Wiley and Sons, 1980. |
[10] |
A. Glankwahmdee, J. S. Liebman and G. L. Hogg,
Unconstrained discrete nonlinear programming, Engineering Optimization, 4 (1979), 95-107.
|
[11] |
P. E. Gill, W. Murray and M. H. Wright, Practical Optimization, Academic Press, London, 1981.
![]() ![]() |
[12] |
J. H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, MI, 1975.
![]() ![]() |
[13] |
A. C. P. Isabel, E. Santo and E. Fernandes,
Heuristics pattern search for bound constrained minimax problems, Computational Science and Its Applications, ICCSA, 6784 (2011), 174-184.
doi: 10.1007/978-3-642-21931-3_15. |
[14] |
R. Jovanovic and M. Tuba,
An ant colony optimization algorithm with improved pheromone correction strategy for the minimum weight vertex cover problem, Applied Soft Computing, 11 (2011), 5360-5366.
|
[15] |
R. Jovanovic and M. Tuba,
Ant colony optimization algorithm with pheromone correction strategy for minimum connected dominating set problem, Computer Science and Information Systems (ComSIS), 10 (2013), 133-149.
doi: 10.2298/CSIS110927038J. |
[16] |
D. Karaboga and B. Basturk,
A powerful and efficient algorithm for numerical function optimization: artificial bee colony (abc) algorithm, Journal of Global Optimization, 39 (2007), 459-471.
doi: 10.1007/s10898-007-9149-x. |
[17] |
J. Kennedy and R. C. Eberhart,
Particle Swarm Optimization, Proceedings of the IEEE International Conference on Neural Networks, 4 (1995), 1942-1948.
|
[18] |
E. C. Laskari, K. E. Parsopoulos and M. N. Vrahatis, Particle swarm optimization for integer programming, Proceedings of the IEEE 2002 Congress on Evolutionary Computation, Honolulu (HI), (2002), 1582–1587. |
[19] |
E. L. Lawler and D. W. Wood,
Branch and bound methods: A survey, Operations Research, 14 (1966), 699-719.
|
[20] |
X. L. Li, Z. J. Shao and J. X. Qian,
An optimizing method based on autonomous animals: Fish-swarm algorithm, System Engineering Theory and Practice, 22 (2003), 32-38.
|
[21] |
G. Liuzzi, S. Lucidi and M. Sciandrone,
A derivative-free algorithm for linearly constrained finite minimax problems, SIAM Journal on Optimization, 16 (2006), 1054-1075.
doi: 10.1137/040615821. |
[22] |
L. Lukan and J. Vlcek, Test problems for nonsmooth unconstrained and linearly constrained optimization, Technical report 798, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic, 2000. |
[23] |
V. M. Manquinho, J. P. Marques Silva, A. L. Oliveira and K. A. Sakallah, Branch and bound algorithms for highly constrained integer programs, Technical Report, Cadence European Laboratories, Portugal, 1997. |
[24] |
S. Mirjalili, S. M. Mirjalili and A. Lewis,
Grey wolf optimizer, Advances in Engineering Software, 69 (2014), 46-61.
|
[25] |
J. A. Nelder and R. Mead,
A simplex method for function minimization, Computer Journal, 7 (1965), 308-313.
doi: 10.1093/comjnl/7.4.308. |
[26] |
G. L. Nemhauser, A. H. G. Rinnooy Kan and M. J. Todd, Handbooks in OR & MS, volume 1. Elsevier, 1989. |
[27] |
K. E. Parsopoulos and M. N. Vrahatis, Unified particle swarm optimization for tackling operations research problems, in Proceeding of IEEE 2005 swarm Intelligence Symposium, Pasadena, USA, (2005), 53-59. |
[28] |
M. K. Passino,
Biomimicry of bacterial foraging for distributed optimization and control, Control Systems, IEEE, 22 (2002), 52-67.
|
[29] |
Y. G. Petalas, K. E. Parsopoulos and M. N. Vrahatis,
Memetic particle swarm optimization, Ann oper Res, 156 (2007), 99-127.
doi: 10.1007/s10479-007-0224-y. |
[30] |
E. Polak, J. O. Royset and R. S. Womersley,
Algorithms with adaptive smoothing for finite minimax problems, Journal of Optimization Theory and Applications, 119 (2003), 459-484.
doi: 10.1023/B:JOTA.0000006685.60019.3e. |
[31] |
S. S. Rao, Engineering Optimization-Theory and Practice, Wiley: New Delhi, 1994. |
[32] |
G. Rudolph, An evolutionary algorithm for integer programming, in Parallel Problem Solving from Nature(eds. Y. Davidor, H-P. Schwefel, and R. Mnner), 3 (1994), 139-148. |
[33] |
E. Sandgen,
Nonlinear integer and discrete programming in mechanical design optimization, Journal of Mechanical Design (ASME), 112 (1990), 223-229.
|
[34] |
H. P. Schwefel, Evolution and Optimum Seeking, New York: Wiley, 1995. |
[35] |
R. Storn and K. Price,
Differential evolution simple and efficient heuristic for global optimization over continuous spaces, J. Glob. Optim., 11 (1997), 341-359.
doi: 10.1023/A:1008202821328. |
[36] |
R. Tang, S. Fong, X. S. Yang and S. Deb, Wolf search algorithm with ephemeral memory, In Digital Information Management (ICDIM), 2012 Seventh International Conference on Digital Information Management, (2012), 165–172. |
[37] |
D. Teodorovic and M. DellOrco. Bee colony optimization cooperative learning approach to complex transportation problems, In Advanced OR and AI Methods in Transportation, Proceedings of 16th MiniEURO Conference and 10th Meeting of EWGT (13-16 September 2005). Poznan: Publishing House of the Polish Operational and System Research, (2005), 51–60. |
[38] |
M. Tuba, N. Bacanin and N. Stanarevic,
Adjusted artificial bee colony (ABC) algorithm for engineering problems, WSEAS Transaction on Computers, 1 (2012), 111-120.
|
[39] |
M. Tuba, M. Subotic and N. Stanarevic,
Performance of a modified cuckoo search algorithm for unconstrained optimization problems, WSEAS Transactions on Systems, 11 (2012), 62-74.
|
[40] |
B. Wilson, A Simplicial Algorithm for Concave Programming, PhD thesis, Harvard University, 1963. |
[41] |
S. Xu,
Smoothing method for minimax problems, Computational Optimization and Applications, 20 (2001), 267-279.
doi: 10.1023/A:1011211101714. |
[42] |
X. S. Yang,
A new metaheuristic bat-inspired algorithm, Nature Inspired Cooperative Strategies for Optimization (NICSO 2010), (2010), 65-74.
|
[43] |
X. S. Yang,
Firefly algorithm, stochastic test functions and design optimisation, International Journal of Bio-Inspired Computation, 2 (2010), 78-84.
|
[44] |
X. S. Yang and S. Deb, Cuckoo search via levy fights, in Nature & Biologically Inspired Computing, NaBIC 2009, World Congress on, IEEE, (2009), 210–214. |
[45] |
Z. Shen, A. Neumaier and M. C. Eiermann,
Solving minimax problems by interval methods, BIT, 30 (1990), 742-751.
doi: 10.1007/BF01933221. |



Parameters | Definitions | Values |
n | Search agents no (population size) | 20 |
Coefficient vector | 2 | |
Coefficient vector | ||
random vectors | [0, 1] | |
Coefficient vector | ||
Number of best solution for final intensification | 1 | |
Maximum number of iterations | 2d -3d |
Parameters | Definitions | Values |
n | Search agents no (population size) | 20 |
Coefficient vector | 2 | |
Coefficient vector | ||
random vectors | [0, 1] | |
Coefficient vector | ||
Number of best solution for final intensification | 1 | |
Maximum number of iterations | 2d -3d |
Test problem | Problem definition |
Problem 1 [32] | |
Problem 2 [32] | |
Problem 3 [10] | |
Problem 4 [10] | |
Problem 5 [10] | |
Problem 6 [32] | |
Problem 7 [10] |
Test problem | Problem definition |
Problem 1 [32] | |
Problem 2 [32] | |
Problem 3 [10] | |
Problem 4 [10] | |
Problem 5 [10] | |
Problem 6 [32] | |
Problem 7 [10] |
Function | Dimension (d) | Bound | Optimal |
FI1 | 5 | [-100 100] | 0 |
FI2 | 5 | [-100 100] | 0 |
FI3 | 5 | [-100 100] | -737 |
FI4 | 2 | [-100 100] | 0 |
FI5 | 4 | [-100 100] | 0 |
FI6 | 2 | [-100 100] | -6 |
FI7 | 2 | [-100 100] | -3833.12 |
Function | Dimension (d) | Bound | Optimal |
FI1 | 5 | [-100 100] | 0 |
FI2 | 5 | [-100 100] | 0 |
FI3 | 5 | [-100 100] | -737 |
FI4 | 2 | [-100 100] | 0 |
FI5 | 4 | [-100 100] | 0 |
FI6 | 2 | [-100 100] | -6 |
FI7 | 2 | [-100 100] | -3833.12 |
Function | Standard | NM | SGWO |
GWO | method | ||
FI1 | 660.4 | 1988.35 | 227.3 |
FI2 | 560.2 | 678.15 | 225.6 |
FI3 | 4920.5 | 819.45 | 701.24 |
FI4 | 2860.3 | 266.14 | 283.5 |
FI5 | 1520.6 | 872.46 | 508.15 |
FI6 | 3760.2 | 254.15 | 364.27 |
FI7 | 1200.3 | 245.47 | 235.16 |
Function | Standard | NM | SGWO |
GWO | method | ||
FI1 | 660.4 | 1988.35 | 227.3 |
FI2 | 560.2 | 678.15 | 225.6 |
FI3 | 4920.5 | 819.45 | 701.24 |
FI4 | 2860.3 | 266.14 | 283.5 |
FI5 | 1520.6 | 872.46 | 508.15 |
FI6 | 3760.2 | 254.15 | 364.27 |
FI7 | 1200.3 | 245.47 | 235.16 |
Function | Algorithm | Min | Max | Mean | St.D | Suc |
FI1 | RWMPSOg | 17,160 | 74,699 | 27,176.3 | 8657 | 50 |
RWMPSOl | 24,870 | 35,265 | 30,923.9 | 2405 | 50 | |
PSOg | 14,000 | 261,100 | 29,435.3 | 42,039 | 34 | |
PSOl | 27,400 | 35,800 | 31,252 | 1818 | 50 | |
SGWO | 225 | 275 | 227.3 | 24.95 | 50 | |
FI2 | RWMPSOg | 252 | 912 | 578.5 | 136.5 | 50 |
RWMPSOl | 369 | 1931 | 773.9 | 285.5 | 50 | |
PSOg | 400 | 1000 | 606.4 | 119 | 50 | |
PSOl | 450 | 1470 | 830.2 | 206 | 50 | |
SGWO | 215 | 250 | 225.6 | 18.52 | 50 | |
FI3 | RWMPSOg | 361 | 41,593 | 6490.6 | 6913 | 50 |
RWMPSOl | 5003 | 15,833 | 9292.6 | 2444 | 50 | |
PSOg | 2150 | 187,000 | 12,681 | 35,067 | 50 | |
PSOl | 4650 | 22,650 | 11,320 | 3803 | 50 | |
SGWO | 715 | 750 | 701.24 | 37.52 | 50 | |
FI4 | RWMPSOg | 76 | 468 | 215 | 97.9 | 50 |
RWMPSOl | 73 | 620 | 218.7 | 115.3 | 50 | |
PSOg | 100 | 620 | 369.6 | 113.2 | 50 | |
PSOl | 120 | 920 | 390 | 134.6 | 50 | |
SGWO | 275 | 290 | 283.5 | 7.63 | 50 | |
FI5 | RWMPSOg | 687 | 2439 | 1521.8 | 360.7 | 50 |
RWMPSOl | 675 | 3863 | 2102.9 | 689.5 | 50 | |
PSOg | 680 | 3440 | 1499 | 513.1 | 43 | |
PSOl | 800 | 3880 | 2472.4 | 637.5 | 50 | |
SGWO | 510 | 540 | 508.15 | 16.07 | 50 | |
FI6 | RWMPSOg | 40 | 238 | 110.9 | 48.6 | 50 |
RWMPSOl | 40 | 235 | 112 | 48.7 | 50 | |
PSOg | 80 | 350 | 204.8 | 62 | 50 | |
PSOl | 70 | 520 | 256 | 107.5 | 50 | |
SGWO | 355 | 370 | 361.6 | 7.63 | 50 | |
FI7 | RWMPSOg | 72 | 620 | 242.7 | 132.2 | 50 |
RWMPSOl | 70 | 573 | 248.9 | 134.4 | 50 | |
PSOg | 100 | 660 | 421.2 | 130.4 | 50 | |
PSOl | 100 | 820 | 466 | 165 | 50 | |
SGWO | 215 | 250 | 235.16 | 18.02 | 50 |
Function | Algorithm | Min | Max | Mean | St.D | Suc |
FI1 | RWMPSOg | 17,160 | 74,699 | 27,176.3 | 8657 | 50 |
RWMPSOl | 24,870 | 35,265 | 30,923.9 | 2405 | 50 | |
PSOg | 14,000 | 261,100 | 29,435.3 | 42,039 | 34 | |
PSOl | 27,400 | 35,800 | 31,252 | 1818 | 50 | |
SGWO | 225 | 275 | 227.3 | 24.95 | 50 | |
FI2 | RWMPSOg | 252 | 912 | 578.5 | 136.5 | 50 |
RWMPSOl | 369 | 1931 | 773.9 | 285.5 | 50 | |
PSOg | 400 | 1000 | 606.4 | 119 | 50 | |
PSOl | 450 | 1470 | 830.2 | 206 | 50 | |
SGWO | 215 | 250 | 225.6 | 18.52 | 50 | |
FI3 | RWMPSOg | 361 | 41,593 | 6490.6 | 6913 | 50 |
RWMPSOl | 5003 | 15,833 | 9292.6 | 2444 | 50 | |
PSOg | 2150 | 187,000 | 12,681 | 35,067 | 50 | |
PSOl | 4650 | 22,650 | 11,320 | 3803 | 50 | |
SGWO | 715 | 750 | 701.24 | 37.52 | 50 | |
FI4 | RWMPSOg | 76 | 468 | 215 | 97.9 | 50 |
RWMPSOl | 73 | 620 | 218.7 | 115.3 | 50 | |
PSOg | 100 | 620 | 369.6 | 113.2 | 50 | |
PSOl | 120 | 920 | 390 | 134.6 | 50 | |
SGWO | 275 | 290 | 283.5 | 7.63 | 50 | |
FI5 | RWMPSOg | 687 | 2439 | 1521.8 | 360.7 | 50 |
RWMPSOl | 675 | 3863 | 2102.9 | 689.5 | 50 | |
PSOg | 680 | 3440 | 1499 | 513.1 | 43 | |
PSOl | 800 | 3880 | 2472.4 | 637.5 | 50 | |
SGWO | 510 | 540 | 508.15 | 16.07 | 50 | |
FI6 | RWMPSOg | 40 | 238 | 110.9 | 48.6 | 50 |
RWMPSOl | 40 | 235 | 112 | 48.7 | 50 | |
PSOg | 80 | 350 | 204.8 | 62 | 50 | |
PSOl | 70 | 520 | 256 | 107.5 | 50 | |
SGWO | 355 | 370 | 361.6 | 7.63 | 50 | |
FI7 | RWMPSOg | 72 | 620 | 242.7 | 132.2 | 50 |
RWMPSOl | 70 | 573 | 248.9 | 134.4 | 50 | |
PSOg | 100 | 660 | 421.2 | 130.4 | 50 | |
PSOl | 100 | 820 | 466 | 165 | 50 | |
SGWO | 215 | 250 | 235.16 | 18.02 | 50 |
Function | GA+NM | DE+NM | PSO+NM | FF+NM | CS+NM | SGWO | |
FI1 | Avg | 1106.56 | 935.48 | 1160.12 | 975.23 | 845.68 | 227.3 |
SD | 457.96 | 256.12 | 423.56 | 235.69 | 115.48 | 24.95 | |
FI2 | Avg | 947.42 | 914.58 | 1125.56 | 911.25 | 984.69 | 225.6 |
SD | 110.07 | 246.24 | 285.46 | 115.48 | 254.89 | 18.52 | |
FI3 | Avg | 2053.43 | 1846.23 | 1915.35 | 1825.23 | 1115.12 | 701.24 |
SD | 41.64 | 115.47 | 235.69 | 245.56 | 158.42 | 37.52 | |
FI4 | Avg | 866.73 | 745.78 | 875.69 | 796.23 | 414.26 | 283.5 |
SD | 284.48 | 125.26 | 145.36 | 175.28 | 118.54 | 7.63 | |
FI5 | Avg | 954.54 | 923.18 | 1115.24 | 960.43 | 1142.58 | 508.15 |
SD | 74.93 | 126.21 | 114.56 | 124.56 | 215.48 | 16.07 | |
FI6 | Avg | 554.35 | 515.24 | 535.46 | 498.75 | 458.49 | 361.6 |
SD | 115.14 | 125.36 | 223.52 | 113.58 | 118.35 | 7.63 | |
FI7 | Avg | 485.14 | 454.36 | 514.12 | 435.47 | 390.78 | 235.16 |
SD | 117.12 | 148.12 | 90.14 | 142.58 | 118.62 | 18.02 |
Function | GA+NM | DE+NM | PSO+NM | FF+NM | CS+NM | SGWO | |
FI1 | Avg | 1106.56 | 935.48 | 1160.12 | 975.23 | 845.68 | 227.3 |
SD | 457.96 | 256.12 | 423.56 | 235.69 | 115.48 | 24.95 | |
FI2 | Avg | 947.42 | 914.58 | 1125.56 | 911.25 | 984.69 | 225.6 |
SD | 110.07 | 246.24 | 285.46 | 115.48 | 254.89 | 18.52 | |
FI3 | Avg | 2053.43 | 1846.23 | 1915.35 | 1825.23 | 1115.12 | 701.24 |
SD | 41.64 | 115.47 | 235.69 | 245.56 | 158.42 | 37.52 | |
FI4 | Avg | 866.73 | 745.78 | 875.69 | 796.23 | 414.26 | 283.5 |
SD | 284.48 | 125.26 | 145.36 | 175.28 | 118.54 | 7.63 | |
FI5 | Avg | 954.54 | 923.18 | 1115.24 | 960.43 | 1142.58 | 508.15 |
SD | 74.93 | 126.21 | 114.56 | 124.56 | 215.48 | 16.07 | |
FI6 | Avg | 554.35 | 515.24 | 535.46 | 498.75 | 458.49 | 361.6 |
SD | 115.14 | 125.36 | 223.52 | 113.58 | 118.35 | 7.63 | |
FI7 | Avg | 485.14 | 454.36 | 514.12 | 435.47 | 390.78 | 235.16 |
SD | 117.12 | 148.12 | 90.14 | 142.58 | 118.62 | 18.02 |
Function | Algorithm | Mean | St.D | Suc |
FI1 | BB | 1167.83 | 659.8 | 30 |
SGWO | 223.15 | 22.35 | 30 | |
FI2 | BB | 139.7 | 102.6 | 30 |
SGWO | 220.26 | 15.26 | 30 | |
FI3 | BB | 4185.5 | 32.8 | 30 |
SGWO | 690.55 | 34.56 | 30 | |
FI4 | BB | 316.9 | 125.4 | 30 |
SGWO | 279.85 | 8.56 | 30 | |
FI5 | BB | 2754 | 1030.1 | 30 |
SGWO | 498.25 | 15.48 | 30 | |
FI6 | BB | 211 | 15 | 30 |
SGWO | 361.75 | 9.45 | 30 | |
FI7 | BB | 358.6 | 14.7 | 30 |
SGWO | 233.45 | 21.45 | 30 |
Function | Algorithm | Mean | St.D | Suc |
FI1 | BB | 1167.83 | 659.8 | 30 |
SGWO | 223.15 | 22.35 | 30 | |
FI2 | BB | 139.7 | 102.6 | 30 |
SGWO | 220.26 | 15.26 | 30 | |
FI3 | BB | 4185.5 | 32.8 | 30 |
SGWO | 690.55 | 34.56 | 30 | |
FI4 | BB | 316.9 | 125.4 | 30 |
SGWO | 279.85 | 8.56 | 30 | |
FI5 | BB | 2754 | 1030.1 | 30 |
SGWO | 498.25 | 15.48 | 30 | |
FI6 | BB | 211 | 15 | 30 |
SGWO | 361.75 | 9.45 | 30 | |
FI7 | BB | 358.6 | 14.7 | 30 |
SGWO | 233.45 | 21.45 | 30 |
Test problem | Problem definition |
Problem 1 [41] | |
Problem 2 [41] | |
Problem 3 [41] | |
Problem 4 [41] | |
Problem 5 [34] | |
Problem 6 [34] | |
Problem 7 [22] | |
Problem 8 [22] | |
Problem 9 [22] | |
Problem 10 [22] |
Test problem | Problem definition |
Problem 1 [41] | |
Problem 2 [41] | |
Problem 3 [41] | |
Problem 4 [41] | |
Problem 5 [34] | |
Problem 6 [34] | |
Problem 7 [22] | |
Problem 8 [22] | |
Problem 9 [22] | |
Problem 10 [22] |
Function | Dimension(d) | Desired error goal |
FM1 | 2 | 1.95222245 |
FM2 | 2 | 2 |
FM3 | 4 | -40.1 |
FM4 | 7 | 247 |
FM5 | 2 | 10−4 |
FM6 | 10 | 10−4 |
FM7 | 2 | 10−4 |
FM8 | 4 | -40.1 |
FM9 | 7 | 680 |
FM10 | 4 | 0.1 |
Function | Dimension(d) | Desired error goal |
FM1 | 2 | 1.95222245 |
FM2 | 2 | 2 |
FM3 | 4 | -40.1 |
FM4 | 7 | 247 |
FM5 | 2 | 10−4 |
FM6 | 10 | 10−4 |
FM7 | 2 | 10−4 |
FM8 | 4 | -40.1 |
FM9 | 7 | 680 |
FM10 | 4 | 0.1 |
Function | Standard | NM | SGWO |
GWO | method | ||
FM1 | 2940.2 | 290.35 | 210.23 |
FM2 | 3740.1 | 286.47 | 195.15 |
FM3 | 1120.2 | 537.46 | 381.75 |
FM4 | 4940.3 | 19,147.15 | 806.45 |
FM5 | 3520.4 | 273.36 | 215.36 |
FM6 | 2080.3 | 18,245.48 | 1602.18 |
FM7 | 1020.4 | 736.14 | 138.62 |
FM8 | 1620.4 | 1652.17 | 373.25 |
FM9 | 3760.5 | 19,857.69 | 942.45 |
FM10 | 1630.4 | 867.26 | 349.46 |
Function | Standard | NM | SGWO |
GWO | method | ||
FM1 | 2940.2 | 290.35 | 210.23 |
FM2 | 3740.1 | 286.47 | 195.15 |
FM3 | 1120.2 | 537.46 | 381.75 |
FM4 | 4940.3 | 19,147.15 | 806.45 |
FM5 | 3520.4 | 273.36 | 215.36 |
FM6 | 2080.3 | 18,245.48 | 1602.18 |
FM7 | 1020.4 | 736.14 | 138.62 |
FM8 | 1620.4 | 1652.17 | 373.25 |
FM9 | 3760.5 | 19,857.69 | 942.45 |
FM10 | 1630.4 | 867.26 | 349.46 |
Algorithm | Problem | Avg | SD | %Suc |
HPS2 | FM1 | 1848.7 | 2619.4 | 99 |
FM2 | 635.8 | 114.3 | 94 | |
FM3 | 141.2 | 28.4 | 37 | |
FM4 | 8948.4 | 5365.4 | 7 | |
FM5 | 772.0 | 60.8 | 100 | |
FM6 | 1809.1 | 2750.3 | 94 | |
FM7 | 4114.7 | 1150.2 | 100 | |
FM8 | - | - | - | |
FM9 | 283.0 | 123.9 | 64 | |
FM10 | 324.1 | 173.1 | 100 | |
UPSOm | FM1 | 1993.8 | 853.7 | 100 |
FM2 | 1775.6 | 241.9 | 100 | |
FM3 | 1670.4 | 530.6 | 100 | |
FM4 | 12,801.5 | 5072.1 | 100 | |
FM5 | 1701.6 | 184.9 | 100 | |
FM6 | 18,294.5 | 2389.4 | 100 | |
FM7 | 3435.5 | 1487.6 | 100 | |
FM8 | 6618.50 | 2597.54 | 100 | |
FM9 | 2128.5 | 597.4 | 100 | |
FM10 | 3332.5 | 1775.4 | 100 | |
RWMPSOg | FM1 | 2415.3 | 1244.2 | 100 |
FM2 | - | - | - | |
FM3 | 3991.3 | 2545.2 | 100 | |
FM4 | 7021.3 | 1241.4 | 100 | |
FM5 | 2947.8 | 257.0 | 100 | |
FM6 | 18,520.1 | 776.9 | 100 | |
FM7 | 1308.8 | 505.5 | 100 | |
FM8 | - | - | - | |
FM9 | - | - | - | |
FM10 | 4404.0 | 3308.9 | 100 | |
SGWO | FM1 | 210.23 | 25.54 | 100 |
FM2 | 195.15 | 36.69 | 100 | |
FM3 | 381.75 | 15.39 | 100 | |
FM4 | 806.45 | 249.55 | 100 | |
FM5 | 215.36 | 75.68 | 100 | |
FM6 | 1602.18 | 425.18 | 100 | |
FM7 | 932.6 | 12.6 | 100 | |
FM8 | 138.62 | 15.23 | 100 | |
FM9 | 942.45 | 55.68 | 100 | |
FM10 | 349.46 | 25.45 | 100 |
Algorithm | Problem | Avg | SD | %Suc |
HPS2 | FM1 | 1848.7 | 2619.4 | 99 |
FM2 | 635.8 | 114.3 | 94 | |
FM3 | 141.2 | 28.4 | 37 | |
FM4 | 8948.4 | 5365.4 | 7 | |
FM5 | 772.0 | 60.8 | 100 | |
FM6 | 1809.1 | 2750.3 | 94 | |
FM7 | 4114.7 | 1150.2 | 100 | |
FM8 | - | - | - | |
FM9 | 283.0 | 123.9 | 64 | |
FM10 | 324.1 | 173.1 | 100 | |
UPSOm | FM1 | 1993.8 | 853.7 | 100 |
FM2 | 1775.6 | 241.9 | 100 | |
FM3 | 1670.4 | 530.6 | 100 | |
FM4 | 12,801.5 | 5072.1 | 100 | |
FM5 | 1701.6 | 184.9 | 100 | |
FM6 | 18,294.5 | 2389.4 | 100 | |
FM7 | 3435.5 | 1487.6 | 100 | |
FM8 | 6618.50 | 2597.54 | 100 | |
FM9 | 2128.5 | 597.4 | 100 | |
FM10 | 3332.5 | 1775.4 | 100 | |
RWMPSOg | FM1 | 2415.3 | 1244.2 | 100 |
FM2 | - | - | - | |
FM3 | 3991.3 | 2545.2 | 100 | |
FM4 | 7021.3 | 1241.4 | 100 | |
FM5 | 2947.8 | 257.0 | 100 | |
FM6 | 18,520.1 | 776.9 | 100 | |
FM7 | 1308.8 | 505.5 | 100 | |
FM8 | - | - | - | |
FM9 | - | - | - | |
FM10 | 4404.0 | 3308.9 | 100 | |
SGWO | FM1 | 210.23 | 25.54 | 100 |
FM2 | 195.15 | 36.69 | 100 | |
FM3 | 381.75 | 15.39 | 100 | |
FM4 | 806.45 | 249.55 | 100 | |
FM5 | 215.36 | 75.68 | 100 | |
FM6 | 1602.18 | 425.18 | 100 | |
FM7 | 932.6 | 12.6 | 100 | |
FM8 | 138.62 | 15.23 | 100 | |
FM9 | 942.45 | 55.68 | 100 | |
FM10 | 349.46 | 25.45 | 100 |
Function | GA+NM | DE+NM | PSO+NM | FF+NM | CS+NM | SGWO | |
FM1 | Avg | 486.25 | 458.47 | 490.78 | 445.42 | 391.16 | 210.23 |
SD | 153.69 | 114.58 | 128.87 | 98.47 | 95.48 | 25.54 | |
FM2 | Avg | 469.58 | 459.28 | 485.46 | 483.47 | 346.58 | 195.15 |
SD | 115.45 | 112.86 | 135.486 | 115.78 | 125.48 | 36.69 | |
FM3 | Avg | 635.48 | 590.46 | 610.76 | 598.48 | 359.42 | 381.75 |
SD | 186.92 | 211.48 | 184.35 | 115.46 | 112.58 | 15.39 | |
FM4 | Avg | 2158.69 | 2214.78 | 1985.46 | 1965.48 | 1846.35 | 806.45 |
SD | 354.76 | 387.45 | 453.84 | 536.44 | 458.75 | 249.55 | |
FM5 | Avg | 476.58 | 436.48 | 469.85 | 456.48 | 315.36 | 215.36 |
SD | 114.79 | 113.58 | 135.48 | 112.47 | 114.56 | 75.68 | |
FM6 | Avg | 5383.49 | 4952.36 | 5148.46 | 4856.24 | 2952.14 | 1602.18 |
SD | 486.58 | 425.85 | 415.68 | 364.58 | 358.45 | 425.18 | |
FM7 | Avg | 487.48 | 495.48 | 496.58 | 468.12 | 295.48 | 138.62 |
SD | 127.85 | 142.36 | 185.26 | 169.35 | 85.34 | 12.6 | |
FM8 | Avg | 2180.35 | 2049.15 | 2185.46 | 1954.15 | 1665.28 | 373.25 |
SD | 487.54 | 475.69 | 519.48 | 413.68 | 98.62 | 15.23 | |
FM9 | Avg | 5982.48 | 5846.48 | 5948.47 | 5634.65 | 3158.46 | 942.45 |
SD | 487.14 | 356.84 | 458.36 | 368.47 | 256.48 | 55.68 | |
FM10 | Avg | 845.71 | 795.26 | 876.29 | 863.45 | 563.58 | 349.46 |
SD | 248.27 | 195.47 | 112.84 | 158.58 | 158.16 | 25.45 |
Function | GA+NM | DE+NM | PSO+NM | FF+NM | CS+NM | SGWO | |
FM1 | Avg | 486.25 | 458.47 | 490.78 | 445.42 | 391.16 | 210.23 |
SD | 153.69 | 114.58 | 128.87 | 98.47 | 95.48 | 25.54 | |
FM2 | Avg | 469.58 | 459.28 | 485.46 | 483.47 | 346.58 | 195.15 |
SD | 115.45 | 112.86 | 135.486 | 115.78 | 125.48 | 36.69 | |
FM3 | Avg | 635.48 | 590.46 | 610.76 | 598.48 | 359.42 | 381.75 |
SD | 186.92 | 211.48 | 184.35 | 115.46 | 112.58 | 15.39 | |
FM4 | Avg | 2158.69 | 2214.78 | 1985.46 | 1965.48 | 1846.35 | 806.45 |
SD | 354.76 | 387.45 | 453.84 | 536.44 | 458.75 | 249.55 | |
FM5 | Avg | 476.58 | 436.48 | 469.85 | 456.48 | 315.36 | 215.36 |
SD | 114.79 | 113.58 | 135.48 | 112.47 | 114.56 | 75.68 | |
FM6 | Avg | 5383.49 | 4952.36 | 5148.46 | 4856.24 | 2952.14 | 1602.18 |
SD | 486.58 | 425.85 | 415.68 | 364.58 | 358.45 | 425.18 | |
FM7 | Avg | 487.48 | 495.48 | 496.58 | 468.12 | 295.48 | 138.62 |
SD | 127.85 | 142.36 | 185.26 | 169.35 | 85.34 | 12.6 | |
FM8 | Avg | 2180.35 | 2049.15 | 2185.46 | 1954.15 | 1665.28 | 373.25 |
SD | 487.54 | 475.69 | 519.48 | 413.68 | 98.62 | 15.23 | |
FM9 | Avg | 5982.48 | 5846.48 | 5948.47 | 5634.65 | 3158.46 | 942.45 |
SD | 487.14 | 356.84 | 458.36 | 368.47 | 256.48 | 55.68 | |
FM10 | Avg | 845.71 | 795.26 | 876.29 | 863.45 | 563.58 | 349.46 |
SD | 248.27 | 195.47 | 112.84 | 158.58 | 158.16 | 25.45 |
Function | Algorithm | Mean | St.D | Suc |
FM1 | SQP | 4044.5 | 8116.6 | 24 |
SGWO | 211.45 | 31.56 | 30 | |
FM2 | SQP | 8035.7 | 9939.9 | 18 |
SGWO | 191.58 | 45.36 | 30 | |
FM3 | SQP | 135.5 | 21.1 | 30 |
SGWO | 385.75 | 27.42 | 30 | |
FM4 | SQP | 20,000 | 0.0 | 0.0 |
SGWO | 825.36 | 250.36 | 30 | |
FM5 | SQP | 140.6 | 38.5 | 30 |
SGWO | 210.45 | 85.65 | 30 | |
FM6 | SQP | 611.6 | 200.6 | 30 |
SGWO | 1648.23 | 512.34 | 30 | |
FM7 | SQP | 15,684.0 | 7302.0 | 10 |
SGWO | 152.34 | 15.48 | 30 | |
FM8 | SQP | 20,000 | 0.0 | 0.0 |
SGWO | 380.26 | 36.89 | 30 | |
FM9 | SQP | 20,000 | 0.0 | 0.0 |
SGWO | 936.48 | 62.35 | 30 | |
FM10 | SQP | 4886.5 | 8488.4 | 22 |
SGWO | 356.89 | 39.85 | 30 |
Function | Algorithm | Mean | St.D | Suc |
FM1 | SQP | 4044.5 | 8116.6 | 24 |
SGWO | 211.45 | 31.56 | 30 | |
FM2 | SQP | 8035.7 | 9939.9 | 18 |
SGWO | 191.58 | 45.36 | 30 | |
FM3 | SQP | 135.5 | 21.1 | 30 |
SGWO | 385.75 | 27.42 | 30 | |
FM4 | SQP | 20,000 | 0.0 | 0.0 |
SGWO | 825.36 | 250.36 | 30 | |
FM5 | SQP | 140.6 | 38.5 | 30 |
SGWO | 210.45 | 85.65 | 30 | |
FM6 | SQP | 611.6 | 200.6 | 30 |
SGWO | 1648.23 | 512.34 | 30 | |
FM7 | SQP | 15,684.0 | 7302.0 | 10 |
SGWO | 152.34 | 15.48 | 30 | |
FM8 | SQP | 20,000 | 0.0 | 0.0 |
SGWO | 380.26 | 36.89 | 30 | |
FM9 | SQP | 20,000 | 0.0 | 0.0 |
SGWO | 936.48 | 62.35 | 30 | |
FM10 | SQP | 4886.5 | 8488.4 | 22 |
SGWO | 356.89 | 39.85 | 30 |
[1] |
C. J. Price. A modified Nelder-Mead barrier method for constrained optimization. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 613-631. doi: 10.3934/naco.2020058 |
[2] |
Rong Hu, Ya-Ping Fang. A parametric simplex algorithm for biobjective piecewise linear programming problems. Journal of Industrial and Management Optimization, 2017, 13 (2) : 573-586. doi: 10.3934/jimo.2016032 |
[3] |
Xiao-Bing Li, Qi-Lin Wang, Zhi Lin. Optimality conditions and duality for minimax fractional programming problems with data uncertainty. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1133-1151. doi: 10.3934/jimo.2018089 |
[4] |
Ye Tian, Cheng Lu. Nonconvex quadratic reformulations and solvable conditions for mixed integer quadratic programming problems. Journal of Industrial and Management Optimization, 2011, 7 (4) : 1027-1039. doi: 10.3934/jimo.2011.7.1027 |
[5] |
Jing Quan, Zhiyou Wu, Guoquan Li. Global optimality conditions for some classes of polynomial integer programming problems. Journal of Industrial and Management Optimization, 2011, 7 (1) : 67-78. doi: 10.3934/jimo.2011.7.67 |
[6] |
Chunming Tang, Jinbao Jian, Guoyin Li. A proximal-projection partial bundle method for convex constrained minimax problems. Journal of Industrial and Management Optimization, 2019, 15 (2) : 757-774. doi: 10.3934/jimo.2018069 |
[7] |
Junxiang Li, Yan Gao, Tao Dai, Chunming Ye, Qiang Su, Jiazhen Huo. Substitution secant/finite difference method to large sparse minimax problems. Journal of Industrial and Management Optimization, 2014, 10 (2) : 637-663. doi: 10.3934/jimo.2014.10.637 |
[8] |
Anurag Jayswal, Ashish Kumar Prasad, Izhar Ahmad. On minimax fractional programming problems involving generalized $(H_p,r)$-invex functions. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1001-1018. doi: 10.3934/jimo.2014.10.1001 |
[9] |
Yongjian Yang, Zhiyou Wu, Fusheng Bai. A filled function method for constrained nonlinear integer programming. Journal of Industrial and Management Optimization, 2008, 4 (2) : 353-362. doi: 10.3934/jimo.2008.4.353 |
[10] |
Yi Xu, Wenyu Sun. A filter successive linear programming method for nonlinear semidefinite programming problems. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 193-206. doi: 10.3934/naco.2012.2.193 |
[11] |
Suxiang He, Yunyun Nie. A class of nonlinear Lagrangian algorithms for minimax problems. Journal of Industrial and Management Optimization, 2013, 9 (1) : 75-97. doi: 10.3934/jimo.2013.9.75 |
[12] |
Jinzhi Wang, Yuduo Zhang. Solving the seepage problems with free surface by mathematical programming method. Numerical Algebra, Control and Optimization, 2015, 5 (4) : 351-357. doi: 10.3934/naco.2015.5.351 |
[13] |
Cheng Ma, Xun Li, Ka-Fai Cedric Yiu, Yongjian Yang, Liansheng Zhang. On an exact penalty function method for semi-infinite programming problems. Journal of Industrial and Management Optimization, 2012, 8 (3) : 705-726. doi: 10.3934/jimo.2012.8.705 |
[14] |
Yanqin Bai, Chuanhao Guo. Doubly nonnegative relaxation method for solving multiple objective quadratic programming problems. Journal of Industrial and Management Optimization, 2014, 10 (2) : 543-556. doi: 10.3934/jimo.2014.10.543 |
[15] |
Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control and Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327 |
[16] |
Zhiguo Feng, Ka-Fai Cedric Yiu. Manifold relaxations for integer programming. Journal of Industrial and Management Optimization, 2014, 10 (2) : 557-566. doi: 10.3934/jimo.2014.10.557 |
[17] |
Yue Lu, Ying-En Ge, Li-Wei Zhang. An alternating direction method for solving a class of inverse semi-definite quadratic programming problems. Journal of Industrial and Management Optimization, 2016, 12 (1) : 317-336. doi: 10.3934/jimo.2016.12.317 |
[18] |
Xiantao Xiao, Liwei Zhang, Jianzhong Zhang. On convergence of augmented Lagrangian method for inverse semi-definite quadratic programming problems. Journal of Industrial and Management Optimization, 2009, 5 (2) : 319-339. doi: 10.3934/jimo.2009.5.319 |
[19] |
Gang Li, Yinghong Xu, Zhenhua Qin. Optimality conditions for composite DC infinite programming problems. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022064 |
[20] |
Xian-Jun Long, Jing Quan. Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 361-370. doi: 10.3934/naco.2011.1.361 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]