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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 selfadaptive projectionauxiliary 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), 862874. doi: 10.1016/j.trb.2007.04.008. 
[2] 
D. Bernstein and S. Gabriel, Solving the nonadditive traffic equilibrium problem, in "Proceedings of the Network Optimization Conference" (eds. P. Pardalos, D. Hearn and W. Hager), Springer, (1997), 72102. 
[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), 139159. 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), 230239. 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), 325333. 
[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), 173205. doi: 10.1023/A:1008705425484. 
[7] 
B. C. Eaves, On the basic theorem of complementarity, Mathematical Programming, 1 (1971), 6875. doi: 10.1007/BF01584073. 
[8] 
F. Facchinei and J. S. Pang, "FiniteDimensional Variational Inequalities and Complementarity Problems," Volumes I and II, Springer Verlag, Berlin, 2003. 
[9] 
M. C. Ferris and J. S. Pang, Engineering and economic applications of complementarity problems, SIAM Review, 39 (1997), 669713. doi: 10.1137/S0036144595285963. 
[10] 
S. Gabriel and D. Bernstein, The traffic equilibrium problem with nonadditive path costs, Transportation Science, 31 (1997), 337348. doi: 10.1287/trsc.31.4.337. 
[11] 
A. A. Goldstein, Convex programming in Hilbert space, Bulletin of the American Mathematical Society, 70 (1964), 709710. doi: 10.1090/S000299041964111782. 
[12] 
D. R. Han and Hong K. Lo, Solving nonadditive traffic assignment problems: A descent method for cocoercive variational inequalities, European Journal of Operational Research, 159 (2004), 529545. doi: 10.1016/S03772217(03)004235. 
[13] 
D. R. Han and W. Y. Sun, A new modified GoldsteinLevitinPolyak projection method for variational inequality problems, Computers and Mathematics with Applications, 47 (2004), 18171825. doi: 10.1016/j.camwa.2003.12.002. 
[14] 
D. R. Han, Inexact operator splitting methods with selfadaptive strategy for variational inequality problems, Journal of Optimization Theory and Applications, 132 (2007), 227243. doi: 10.1007/s1095700690605. 
[15] 
D. R. Han, W. Xu and H. Yang, An operator splitting method for variational inequalities with partially unknown mappings, Numerische Mathematik, 111 (2008), 207237. doi: 10.1007/s0021100801817. 
[16] 
P. T. Harker and J. S. Pang, Finitedimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Mathematical Programming, 48 (1990), 161220. 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), 6976. 
[18] 
B. S. He, H. Yang, Q. Meng and D. R. Han, Modified GoldsteinLevitinPolyak projection method for asymmetric strongly monotone variational inequalities, Journal of Optimization Theory and Applications, 112 (2002), 129143. doi: 10.1023/A:1013048729944. 
[19] 
B. S. He, L. Z. Liao and S. L. Wang, Selfadaptive operator splitting methods for monotone variational inequalities, Numerische Mathematik, 94 (2003), 715737. 
[20] 
B. S. He, X. H. Liu, T. Wu and X. Z. He, Selfadaptive projection method for cocoercive variational inequalities, European Journal of Operational Research, 196 (2009), 4348. 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), 747756. 
[22] 
E. N. Khobotov, Modification of the extragradient method for solving variational inequalities and certain optimization problems, USSR, Computational Mathematics and Mathematical Physics, 27 (1987), 120127. doi: 10.1016/00415553(87)900589. 
[23] 
E. S. Levitin and B. T. Polyak, Constrained minimization problems, USSR. Computational Mathematics and Mathematical Physics, 6 (1966), 150. doi: 10.1016/00415553(66)901145. 
[24] 
H. Lo, A dynamic traffic assignment formulation that encapsulates the Cell Transmission Model, in "Transportation and Traffic Theory" (ed. A. Cedar), Elsevier Science, (1999), 327350. 
[25] 
H. Lo and A. Chen, Reformulating the traffic equilibrium problem via a smooth gap function, Mathematical and Computer Modeling, 31 (2000), 179195. doi: 10.1016/S08957177(99)002319. 
[26] 
H. Lo and A. Chen, Traffic equilibrium problem with routespecific costs: Formulation and algorithms, Transportation Research Part B, 34 (2000), 493513. doi: 10.1016/S01912615(99)000351. 
[27] 
A. Nagurney, "Network Economics: A Variational Inequality Approach," Kluwer Academics Publishers, Dordrecht, Boston, London, 1993. 
[28] 
K. Scott and D. Bernstein, Solving a traffic equilibrium problem when path costs are nonadditive, Paper presented at "The 78th Annual Meeting of the Transportation Research Board", Washington, DC, January 1999. 
[29] 
K. Taji, M. Fukushima and T. Ibaraki, A globally convergent Newton method for solving strongly monotone variational inequalities, Mathematical Programming, 58 (1993), 369383. doi: 10.1007/BF01581276. 
[30] 
J. G. Wardrop, Some theoretical aspects of road traffic research, in "Proceedings of Institute of Civil Engineers, Pt. II", 1 (1952), 325378. 
[31] 
H. Yang, Q. Meng and D. H. Lee, Trialanderror implementation of marginalcost pricing on networks in the absence of demand functions, Transportation Research Part B, 38 (2004), 477493. doi: 10.1016/S01912615(03)000778. 
[32] 
D. L. Zhu and P. Marcotte, Cocoercivity and its role in the convergence of iterative schemes for solving variational inequalities, SIAM Journal on Optimization, 6 (1996), 714726. 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), 453456. doi: 10.7153/mia0745. 
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), 862874. doi: 10.1016/j.trb.2007.04.008. 
[2] 
D. Bernstein and S. Gabriel, Solving the nonadditive traffic equilibrium problem, in "Proceedings of the Network Optimization Conference" (eds. P. Pardalos, D. Hearn and W. Hager), Springer, (1997), 72102. 
[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), 139159. 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), 230239. 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), 325333. 
[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), 173205. doi: 10.1023/A:1008705425484. 
[7] 
B. C. Eaves, On the basic theorem of complementarity, Mathematical Programming, 1 (1971), 6875. doi: 10.1007/BF01584073. 
[8] 
F. Facchinei and J. S. Pang, "FiniteDimensional Variational Inequalities and Complementarity Problems," Volumes I and II, Springer Verlag, Berlin, 2003. 
[9] 
M. C. Ferris and J. S. Pang, Engineering and economic applications of complementarity problems, SIAM Review, 39 (1997), 669713. doi: 10.1137/S0036144595285963. 
[10] 
S. Gabriel and D. Bernstein, The traffic equilibrium problem with nonadditive path costs, Transportation Science, 31 (1997), 337348. doi: 10.1287/trsc.31.4.337. 
[11] 
A. A. Goldstein, Convex programming in Hilbert space, Bulletin of the American Mathematical Society, 70 (1964), 709710. doi: 10.1090/S000299041964111782. 
[12] 
D. R. Han and Hong K. Lo, Solving nonadditive traffic assignment problems: A descent method for cocoercive variational inequalities, European Journal of Operational Research, 159 (2004), 529545. doi: 10.1016/S03772217(03)004235. 
[13] 
D. R. Han and W. Y. Sun, A new modified GoldsteinLevitinPolyak projection method for variational inequality problems, Computers and Mathematics with Applications, 47 (2004), 18171825. doi: 10.1016/j.camwa.2003.12.002. 
[14] 
D. R. Han, Inexact operator splitting methods with selfadaptive strategy for variational inequality problems, Journal of Optimization Theory and Applications, 132 (2007), 227243. doi: 10.1007/s1095700690605. 
[15] 
D. R. Han, W. Xu and H. Yang, An operator splitting method for variational inequalities with partially unknown mappings, Numerische Mathematik, 111 (2008), 207237. doi: 10.1007/s0021100801817. 
[16] 
P. T. Harker and J. S. Pang, Finitedimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Mathematical Programming, 48 (1990), 161220. 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), 6976. 
[18] 
B. S. He, H. Yang, Q. Meng and D. R. Han, Modified GoldsteinLevitinPolyak projection method for asymmetric strongly monotone variational inequalities, Journal of Optimization Theory and Applications, 112 (2002), 129143. doi: 10.1023/A:1013048729944. 
[19] 
B. S. He, L. Z. Liao and S. L. Wang, Selfadaptive operator splitting methods for monotone variational inequalities, Numerische Mathematik, 94 (2003), 715737. 
[20] 
B. S. He, X. H. Liu, T. Wu and X. Z. He, Selfadaptive projection method for cocoercive variational inequalities, European Journal of Operational Research, 196 (2009), 4348. 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), 747756. 
[22] 
E. N. Khobotov, Modification of the extragradient method for solving variational inequalities and certain optimization problems, USSR, Computational Mathematics and Mathematical Physics, 27 (1987), 120127. doi: 10.1016/00415553(87)900589. 
[23] 
E. S. Levitin and B. T. Polyak, Constrained minimization problems, USSR. Computational Mathematics and Mathematical Physics, 6 (1966), 150. doi: 10.1016/00415553(66)901145. 
[24] 
H. Lo, A dynamic traffic assignment formulation that encapsulates the Cell Transmission Model, in "Transportation and Traffic Theory" (ed. A. Cedar), Elsevier Science, (1999), 327350. 
[25] 
H. Lo and A. Chen, Reformulating the traffic equilibrium problem via a smooth gap function, Mathematical and Computer Modeling, 31 (2000), 179195. doi: 10.1016/S08957177(99)002319. 
[26] 
H. Lo and A. Chen, Traffic equilibrium problem with routespecific costs: Formulation and algorithms, Transportation Research Part B, 34 (2000), 493513. doi: 10.1016/S01912615(99)000351. 
[27] 
A. Nagurney, "Network Economics: A Variational Inequality Approach," Kluwer Academics Publishers, Dordrecht, Boston, London, 1993. 
[28] 
K. Scott and D. Bernstein, Solving a traffic equilibrium problem when path costs are nonadditive, Paper presented at "The 78th Annual Meeting of the Transportation Research Board", Washington, DC, January 1999. 
[29] 
K. Taji, M. Fukushima and T. Ibaraki, A globally convergent Newton method for solving strongly monotone variational inequalities, Mathematical Programming, 58 (1993), 369383. doi: 10.1007/BF01581276. 
[30] 
J. G. Wardrop, Some theoretical aspects of road traffic research, in "Proceedings of Institute of Civil Engineers, Pt. II", 1 (1952), 325378. 
[31] 
H. Yang, Q. Meng and D. H. Lee, Trialanderror implementation of marginalcost pricing on networks in the absence of demand functions, Transportation Research Part B, 38 (2004), 477493. doi: 10.1016/S01912615(03)000778. 
[32] 
D. L. Zhu and P. Marcotte, Cocoercivity and its role in the convergence of iterative schemes for solving variational inequalities, SIAM Journal on Optimization, 6 (1996), 714726. 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), 453456. doi: 10.7153/mia0745. 
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