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An outcome space algorithm for minimizing the product of two convex functions over a convex set
Solving nonadditive traffic assignment problems: A self-adaptive projection-auxiliary problem method for variational inequalities
1. | School of Computer Sciences, Nanjing Normal University, Nanjing 210097 |
2. | School of Mathematical Science, Nanjing Normal University, Nanjing 210046 |
3. | School of Mathematical Sciences, Key Laboratory for NSLSCS of Jiangsu Province, Nanjing Normal University, Nanjing 210046, China |
4. | Department of Civil Engineering, The Hong Kong University of Science and Technology, Hong Kong |
References:
[1] |
R. P. Agdeppa, N. Yamashita and M. Fukushima, The traffic equilibrium problem with nonadditive costs and its monotone mixed complementarity problem formulation,, Transportation Research Part B, 41 (2007), 862.
doi: 10.1016/j.trb.2007.04.008. |
[2] |
D. Bernstein and S. Gabriel, Solving the non-additive traffic equilibrium problem,, in, (1997), 72.
|
[3] |
D. P. Bertsekas and E. M. Gafni, Projection method for variational inequalities with application to the traffic assignment problem,, Mathematical Programming Study, 17 (1982), 139.
doi: 10.1007/BFb0120965. |
[4] |
A. Bnouhachem, M. H. Xu, X. L. Fu and Z. H. Sheng, Modified extragradient methods for solving variational inequalities,, Computers and Mathematics with Applications, 57 (2009), 230.
doi: 10.1016/j.camwa.2008.10.065. |
[5] |
G. Cohen, Auxiliary problem principle extended to variational inequalities,, Journal of Optimization Theory and Applications, 59 (1988), 325.
|
[6] |
T. De Luca, F. Facchinei and C. Kanzow, A theoretical and numerical comparison of some semismooth algorithms for complementarity problems,, Computational Optimization and Applications, 16 (2000), 173.
doi: 10.1023/A:1008705425484. |
[7] |
B. C. Eaves, On the basic theorem of complementarity,, Mathematical Programming, 1 (1971), 68.
doi: 10.1007/BF01584073. |
[8] |
F. Facchinei and J. S. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems,", Volumes I and II, (2003).
|
[9] |
M. C. Ferris and J. S. Pang, Engineering and economic applications of complementarity problems,, SIAM Review, 39 (1997), 669.
doi: 10.1137/S0036144595285963. |
[10] |
S. Gabriel and D. Bernstein, The traffic equilibrium problem with non-additive path costs,, Transportation Science, 31 (1997), 337.
doi: 10.1287/trsc.31.4.337. |
[11] |
A. A. Goldstein, Convex programming in Hilbert space,, Bulletin of the American Mathematical Society, 70 (1964), 709.
doi: 10.1090/S0002-9904-1964-11178-2. |
[12] |
D. R. Han and Hong K. Lo, Solving non-additive traffic assignment problems: A descent method for co-coercive variational inequalities,, European Journal of Operational Research, 159 (2004), 529.
doi: 10.1016/S0377-2217(03)00423-5. |
[13] |
D. R. Han and W. Y. Sun, A new modified Goldstein-Levitin-Polyak projection method for variational inequality problems,, Computers and Mathematics with Applications, 47 (2004), 1817.
doi: 10.1016/j.camwa.2003.12.002. |
[14] |
D. R. Han, Inexact operator splitting methods with self-adaptive strategy for variational inequality problems,, Journal of Optimization Theory and Applications, 132 (2007), 227.
doi: 10.1007/s10957-006-9060-5. |
[15] |
D. R. Han, W. Xu and H. Yang, An operator splitting method for variational inequalities with partially unknown mappings,, Numerische Mathematik, 111 (2008), 207.
doi: 10.1007/s00211-008-0181-7. |
[16] |
P. T. Harker and J. S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications,, Mathematical Programming, 48 (1990), 161.
doi: 10.1007/BF01582255. |
[17] |
B. S. He, A class of projection and contraction methods for monotone variational inequalities,, Applied Mathematics and Optimization, 35 (1997), 69.
|
[18] |
B. S. He, H. Yang, Q. Meng and D. R. Han, Modified Goldstein-Levitin-Polyak projection method for asymmetric strongly monotone variational inequalities,, Journal of Optimization Theory and Applications, 112 (2002), 129.
doi: 10.1023/A:1013048729944. |
[19] |
B. S. He, L. Z. Liao and S. L. Wang, Self-adaptive operator splitting methods for monotone variational inequalities,, Numerische Mathematik, 94 (2003), 715.
|
[20] |
B. S. He, X. H. Liu, T. Wu and X. Z. He, Self-adaptive projection method for co-coercive variational inequalities,, European Journal of Operational Research, 196 (2009), 43.
doi: 10.1016/j.ejor.2008.03.004. |
[21] |
G. M. Korpolevich, The extragradient method for finding saddle points and other problems,, Matecon, 12 (1976), 747.
|
[22] |
E. N. Khobotov, Modification of the extragradient method for solving variational inequalities and certain optimization problems,, USSR, 27 (1987), 120.
doi: 10.1016/0041-5553(87)90058-9. |
[23] |
E. S. Levitin and B. T. Polyak, Constrained minimization problems,, USSR. Computational Mathematics and Mathematical Physics, 6 (1966), 1.
doi: 10.1016/0041-5553(66)90114-5. |
[24] |
H. Lo, A dynamic traffic assignment formulation that encapsulates the Cell Transmission Model,, in, (1999), 327. Google Scholar |
[25] |
H. Lo and A. Chen, Reformulating the traffic equilibrium problem via a smooth gap function,, Mathematical and Computer Modeling, 31 (2000), 179.
doi: 10.1016/S0895-7177(99)00231-9. |
[26] |
H. Lo and A. Chen, Traffic equilibrium problem with route-specific costs: Formulation and algorithms,, Transportation Research Part B, 34 (2000), 493.
doi: 10.1016/S0191-2615(99)00035-1. |
[27] |
A. Nagurney, "Network Economics: A Variational Inequality Approach,", Kluwer Academics Publishers, (1993).
|
[28] |
K. Scott and D. Bernstein, Solving a traffic equilibrium problem when path costs are non-additive,, Paper presented at, (1999). Google Scholar |
[29] |
K. Taji, M. Fukushima and T. Ibaraki, A globally convergent Newton method for solving strongly monotone variational inequalities,, Mathematical Programming, 58 (1993), 369.
doi: 10.1007/BF01581276. |
[30] |
J. G. Wardrop, Some theoretical aspects of road traffic research,, in, 1 (1952), 325. Google Scholar |
[31] |
H. Yang, Q. Meng and D. H. Lee, Trial-and-error implementation of marginal-cost pricing on networks in the absence of demand functions,, Transportation Research Part B, 38 (2004), 477.
doi: 10.1016/S0191-2615(03)00077-8. |
[32] |
D. L. Zhu and P. Marcotte, Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities,, SIAM Journal on Optimization, 6 (1996), 714.
doi: 10.1137/S1052623494250415. |
[33] |
T. Zhu and Z. G. Yu, A simple proof for some important properties of the projection mapping,, Mathematical Inequalities and Applicatrions, 7 (2004), 453.
doi: 10.7153/mia-07-45. |
show all references
References:
[1] |
R. P. Agdeppa, N. Yamashita and M. Fukushima, The traffic equilibrium problem with nonadditive costs and its monotone mixed complementarity problem formulation,, Transportation Research Part B, 41 (2007), 862.
doi: 10.1016/j.trb.2007.04.008. |
[2] |
D. Bernstein and S. Gabriel, Solving the non-additive traffic equilibrium problem,, in, (1997), 72.
|
[3] |
D. P. Bertsekas and E. M. Gafni, Projection method for variational inequalities with application to the traffic assignment problem,, Mathematical Programming Study, 17 (1982), 139.
doi: 10.1007/BFb0120965. |
[4] |
A. Bnouhachem, M. H. Xu, X. L. Fu and Z. H. Sheng, Modified extragradient methods for solving variational inequalities,, Computers and Mathematics with Applications, 57 (2009), 230.
doi: 10.1016/j.camwa.2008.10.065. |
[5] |
G. Cohen, Auxiliary problem principle extended to variational inequalities,, Journal of Optimization Theory and Applications, 59 (1988), 325.
|
[6] |
T. De Luca, F. Facchinei and C. Kanzow, A theoretical and numerical comparison of some semismooth algorithms for complementarity problems,, Computational Optimization and Applications, 16 (2000), 173.
doi: 10.1023/A:1008705425484. |
[7] |
B. C. Eaves, On the basic theorem of complementarity,, Mathematical Programming, 1 (1971), 68.
doi: 10.1007/BF01584073. |
[8] |
F. Facchinei and J. S. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems,", Volumes I and II, (2003).
|
[9] |
M. C. Ferris and J. S. Pang, Engineering and economic applications of complementarity problems,, SIAM Review, 39 (1997), 669.
doi: 10.1137/S0036144595285963. |
[10] |
S. Gabriel and D. Bernstein, The traffic equilibrium problem with non-additive path costs,, Transportation Science, 31 (1997), 337.
doi: 10.1287/trsc.31.4.337. |
[11] |
A. A. Goldstein, Convex programming in Hilbert space,, Bulletin of the American Mathematical Society, 70 (1964), 709.
doi: 10.1090/S0002-9904-1964-11178-2. |
[12] |
D. R. Han and Hong K. Lo, Solving non-additive traffic assignment problems: A descent method for co-coercive variational inequalities,, European Journal of Operational Research, 159 (2004), 529.
doi: 10.1016/S0377-2217(03)00423-5. |
[13] |
D. R. Han and W. Y. Sun, A new modified Goldstein-Levitin-Polyak projection method for variational inequality problems,, Computers and Mathematics with Applications, 47 (2004), 1817.
doi: 10.1016/j.camwa.2003.12.002. |
[14] |
D. R. Han, Inexact operator splitting methods with self-adaptive strategy for variational inequality problems,, Journal of Optimization Theory and Applications, 132 (2007), 227.
doi: 10.1007/s10957-006-9060-5. |
[15] |
D. R. Han, W. Xu and H. Yang, An operator splitting method for variational inequalities with partially unknown mappings,, Numerische Mathematik, 111 (2008), 207.
doi: 10.1007/s00211-008-0181-7. |
[16] |
P. T. Harker and J. S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications,, Mathematical Programming, 48 (1990), 161.
doi: 10.1007/BF01582255. |
[17] |
B. S. He, A class of projection and contraction methods for monotone variational inequalities,, Applied Mathematics and Optimization, 35 (1997), 69.
|
[18] |
B. S. He, H. Yang, Q. Meng and D. R. Han, Modified Goldstein-Levitin-Polyak projection method for asymmetric strongly monotone variational inequalities,, Journal of Optimization Theory and Applications, 112 (2002), 129.
doi: 10.1023/A:1013048729944. |
[19] |
B. S. He, L. Z. Liao and S. L. Wang, Self-adaptive operator splitting methods for monotone variational inequalities,, Numerische Mathematik, 94 (2003), 715.
|
[20] |
B. S. He, X. H. Liu, T. Wu and X. Z. He, Self-adaptive projection method for co-coercive variational inequalities,, European Journal of Operational Research, 196 (2009), 43.
doi: 10.1016/j.ejor.2008.03.004. |
[21] |
G. M. Korpolevich, The extragradient method for finding saddle points and other problems,, Matecon, 12 (1976), 747.
|
[22] |
E. N. Khobotov, Modification of the extragradient method for solving variational inequalities and certain optimization problems,, USSR, 27 (1987), 120.
doi: 10.1016/0041-5553(87)90058-9. |
[23] |
E. S. Levitin and B. T. Polyak, Constrained minimization problems,, USSR. Computational Mathematics and Mathematical Physics, 6 (1966), 1.
doi: 10.1016/0041-5553(66)90114-5. |
[24] |
H. Lo, A dynamic traffic assignment formulation that encapsulates the Cell Transmission Model,, in, (1999), 327. Google Scholar |
[25] |
H. Lo and A. Chen, Reformulating the traffic equilibrium problem via a smooth gap function,, Mathematical and Computer Modeling, 31 (2000), 179.
doi: 10.1016/S0895-7177(99)00231-9. |
[26] |
H. Lo and A. Chen, Traffic equilibrium problem with route-specific costs: Formulation and algorithms,, Transportation Research Part B, 34 (2000), 493.
doi: 10.1016/S0191-2615(99)00035-1. |
[27] |
A. Nagurney, "Network Economics: A Variational Inequality Approach,", Kluwer Academics Publishers, (1993).
|
[28] |
K. Scott and D. Bernstein, Solving a traffic equilibrium problem when path costs are non-additive,, Paper presented at, (1999). Google Scholar |
[29] |
K. Taji, M. Fukushima and T. Ibaraki, A globally convergent Newton method for solving strongly monotone variational inequalities,, Mathematical Programming, 58 (1993), 369.
doi: 10.1007/BF01581276. |
[30] |
J. G. Wardrop, Some theoretical aspects of road traffic research,, in, 1 (1952), 325. Google Scholar |
[31] |
H. Yang, Q. Meng and D. H. Lee, Trial-and-error implementation of marginal-cost pricing on networks in the absence of demand functions,, Transportation Research Part B, 38 (2004), 477.
doi: 10.1016/S0191-2615(03)00077-8. |
[32] |
D. L. Zhu and P. Marcotte, Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities,, SIAM Journal on Optimization, 6 (1996), 714.
doi: 10.1137/S1052623494250415. |
[33] |
T. Zhu and Z. G. Yu, A simple proof for some important properties of the projection mapping,, Mathematical Inequalities and Applicatrions, 7 (2004), 453.
doi: 10.7153/mia-07-45. |
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