American Institute of Mathematical Sciences

Advanced
January  2022, 18(1): 593-611. doi: 10.3934/jimo.2020170

Cooperation in traffic network problems via evolutionary split variational inequalities

Shipra Singh 1, , Aviv Gibali 2,3, and  Xiaolong Qin 4,,

1. 

Department of Mathematics, Technion - Israel Institute of Technology, Haifa, 3200003, Israel
2. 

Department of Mathematics, ORT Braude College, Karmiel, 2161002, Israel
3. 

The Center for Mathematics and Scientific Computation, University of Haifa, Mt. Carmel, Haifa, 3498838, Israel
4. 

Department of Mathematics, Zhejiang Normal University, Zhejiang, China

* Corresponding author: Xiaolong Qin

Received  August 2020 Revised  September 2020 Published  January 2022 Early access  November 2020

In this paper, we construct an evolutionary (time-dependent) split variational inequality problem and show how to reformulate equilibria of the dynamic traffic network models of two cities as such problem. We also establish existence result for the proposed model. Primary numerical results of equilibria illustrate the validity and applicability of our results.

Keywords: Evolutionary split variational inequality problem, Wardrop condition, traffic equilibrium problems, equilibrium solutions, projected dynamical system.
Mathematics Subject Classification: Primary:90C33, 90C39, 49J40.
Citation: Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2022, 18 (1) : 593-611. doi: 10.3934/jimo.2020170
References:
[1]

Q. H. Ansari and M. Rezaei, Existence results for Stampacchia and Minty type vector variational inequalities, Optimization, 59 (2010), 1053-1065.  doi: 10.1080/02331930903395725.  Google Scholar

[2]

D. Aussel and J. Cotrina, Existence of time-dependent traffic equilibria, Appl. Anal., 91 (2012), 1775-1791.  doi: 10.1080/00036811.2012.692364.  Google Scholar

[3]

D. Aussel, R. Gupta and A. Mehra, Evolutionary variational inequality formulation of the generalized Nash equilibrium problem, J. Optim. Theory Appl., 169 (2016), 74-90.  doi: 10.1007/s10957-015-0859-9.  Google Scholar

[4]

A. Barbagallo, Degenerate time-dependent variational inequalities with applications to traffic equilibrium problems, Comput. Methods Appl. Math., 6 (2006), 117-133.  doi: 10.2478/cmam-2006-0006.  Google Scholar

[5]

A. Barbagallo and M.-G. Cojocaru, Dynamic equilibrium formulation of the oligopolistic market problem, Math. Comput. Modelling, 49 (2009), 966-976.  doi: 10.1016/j.mcm.2008.02.003.  Google Scholar

[6]

A. Barbagallo, P. Daniele and A. Maugeri, Variational formulation for a general dynamic financial equilibrium problem: Balance law and liability formula, Nonlinear Anal., 75 (2012), 1104-1123.  doi: 10.1016/j.na.2010.10.013.  Google Scholar

[7]

H. Brezis, Inéquations d'évolution abstraites, C. R. Acad. Sci. Paris Sér. A-B, 264 (1967), A732-A735.  Google Scholar

[8]

C. Byrne, Y. Censor, A. Gibali and S. Reich, The split common null point problem, J. Nonlinear Convex Anal., 13 (2012), 759-775.   Google Scholar

[9]

L.-C. Ceng and S. Huang, Existence theorems for generalized vector variational inequalities with a variable ordering relation, J. Global Optim., 46 (2010), 521-535.  doi: 10.1007/s10898-009-9436-9.  Google Scholar

[10]

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.  Google Scholar

[11]

Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017.  Google Scholar

[12]

Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 59 (2012), 301-323.  doi: 10.1007/s11075-011-9490-5.  Google Scholar

[13]

C. Ciarcià and P. Daniele, New existence theorems for quasi-variational inequalities and applications to financial models, European J. Oper. Res., 251 (2016), 288-299.  doi: 10.1016/j.ejor.2015.11.013.  Google Scholar

[14]

M. Chen and C. Huang, A power penalty method for a class of linearly constrained variational inequality, J. Ind. Manag. Optim., 14 (2018), 1381-1396.  doi: 10.3934/jimo.2018012.  Google Scholar

[15]

M. G. Cojocaru, P. Daniele and A. Nagurney, Projected dynamical systems and evolutionary variational inequalities via Hilbert spaces with applications, J. Optim. Theory Appl., 127 (2005), 549-563.  doi: 10.1007/s10957-005-7502-0.  Google Scholar

[16]

M.-G. Cojocaru and L. B. Jonker, Existence of solutions to projected differential equations in Hilbert spaces, Proc. Amer. Math. Soc., 132 (2004), 183-193.  doi: 10.1090/S0002-9939-03-07015-1.  Google Scholar

[17]

S. Dafermos, Traffic equilibrium and variational inequalities, Transportation Sci., 14 (1980), 42-54.  doi: 10.1287/trsc.14.1.42.  Google Scholar

[18]

P. Daniele, Time-dependent spatial price equilibrium problem: Existence and stability results for the quantity formulation model, J. Global Optim., 28 (2004), 283-295.  doi: 10.1023/B:JOGO.0000026449.29735.3c.  Google Scholar

[19]

P. Daniele, A. Maugeri and W. Oettli, Time-dependent traffic equilibria, J. Optim. Theory Appl., 103 (1999), 543-555.  doi: 10.1023/A:1021779823196.  Google Scholar

[20]

P. Dupuis and H. Ishii, On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications, Stochastics Stochastics Rep., 35 (1991), 31-62.  doi: 10.1080/17442509108833688.  Google Scholar

[21]

P. Dupuis and A. Nagurney, Dynamical systems and variational inequalities, Ann. Oper. Res., 44 (1993), 9-42.  doi: 10.1007/BF02073589.  Google Scholar

[22]

K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann., 266 (1984), 519-537.  doi: 10.1007/BF01458545.  Google Scholar

[23]

G. Fichera, Problemi elastostatici con vincoli unilaterali: Il problema di Signorini con ambique condizioni al contorno, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia (8), 7 (1963/64), 91-140.   Google Scholar

[24]

G. Fichera, Sul problema elastostatico di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), 34 (1963), 138-142.   Google Scholar

[25]

S. Giuffrè, G. Idone and S. Pia, Some classes of projected dynamical systems in Banach spaces and variational inequalities, J. Global Optim., 40 (2008), 119-128.  doi: 10.1007/s10898-007-9173-x.  Google Scholar

[26]

S. Lawphongpanich and D. W. Hearn, Simplical decomposition of the asymmetric traffic assignment problem, Transportation Res. Part B, 18 (1984), 123-133.  doi: 10.1016/0191-2615(84)90026-2.  Google Scholar

[27]

J.-L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493-519.  doi: 10.1002/cpa.3160200302.  Google Scholar

[28]

S.-Y. Matsushita and L. Xu, On finite convergence of iterative methods for variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 161 (2014), 701-715.  doi: 10.1007/s10957-013-0460-z.  Google Scholar

[29]

A. Moudafi, The split common fixed-point problem for demicontractive mappings, Inverse Problems, 26 (2010), 6pp. doi: 10.1088/0266-5611/26/5/055007.  Google Scholar

[30]

A. Nagurney, D. Parkes and P. Daniele, The Internet, evolutionary variational inequalities, and the time-dependent Braess paradox, Comput. Manag. Sci., 4 (2007), 355-375.  doi: 10.1007/s10287-006-0027-7.  Google Scholar

[31]

B. Panicucci, M. Pappalardo and M. Passacantando, A path-based double projection method for solving the asymmetric traffic network equilibrium problem, Optim. Lett., 1 (2007), 171-185.  doi: 10.1007/s11590-006-0002-9.  Google Scholar

[32]

F. Raciti, Equilibrium conditions and vector variational inequalities: A complex relation, J. Global Optim., 40 (2008), 353-360.  doi: 10.1007/s10898-007-9202-9.  Google Scholar

[33]

L. Scrimali and C. Mirabella, Cooperation in pollution control problems via evolutionary variational inequalities, J. Global Optim., 70 (2018), 455-476.  doi: 10.1007/s10898-017-0580-3.  Google Scholar

[34]

Y. Shehu and O. Iyiola, On a modified extragradient method for variational inequality problem with application to industrial electricity production, J. Ind. Manag. Optim., 15 (2019), 319-342.  doi: 10.3934/jimo.2018045.  Google Scholar

[35]

M. J. Smith, The existence, uniqueness and stability of traffic equilibria, Transportation Res., 13 (1979), 295-304.  doi: 10.1016/0191-2615(79)90022-5.  Google Scholar

[36]

G. Stampacchia, Formes bilinéaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris, 258 (1964), 4413-4416.   Google Scholar

[37]

X. K. Sun and S. J. Li, Duality and gap function for generalized multivalued $\epsilon$-vector variational inequality, Appl. Anal., 92 (2013), 482-492.  doi: 10.1080/00036811.2011.628940.  Google Scholar

[38]

J. Yang, Dynamic power price problem: An inverse variational inequality approach, J. Ind. Manag. Optim., 4 (2008), 673-684.  doi: 10.3934/jimo.2008.4.673.  Google Scholar

show all references

References:
[1]

Q. H. Ansari and M. Rezaei, Existence results for Stampacchia and Minty type vector variational inequalities, Optimization, 59 (2010), 1053-1065.  doi: 10.1080/02331930903395725.  Google Scholar

[2]

D. Aussel and J. Cotrina, Existence of time-dependent traffic equilibria, Appl. Anal., 91 (2012), 1775-1791.  doi: 10.1080/00036811.2012.692364.  Google Scholar

[3]

D. Aussel, R. Gupta and A. Mehra, Evolutionary variational inequality formulation of the generalized Nash equilibrium problem, J. Optim. Theory Appl., 169 (2016), 74-90.  doi: 10.1007/s10957-015-0859-9.  Google Scholar

[4]

A. Barbagallo, Degenerate time-dependent variational inequalities with applications to traffic equilibrium problems, Comput. Methods Appl. Math., 6 (2006), 117-133.  doi: 10.2478/cmam-2006-0006.  Google Scholar

[5]

A. Barbagallo and M.-G. Cojocaru, Dynamic equilibrium formulation of the oligopolistic market problem, Math. Comput. Modelling, 49 (2009), 966-976.  doi: 10.1016/j.mcm.2008.02.003.  Google Scholar

[6]

A. Barbagallo, P. Daniele and A. Maugeri, Variational formulation for a general dynamic financial equilibrium problem: Balance law and liability formula, Nonlinear Anal., 75 (2012), 1104-1123.  doi: 10.1016/j.na.2010.10.013.  Google Scholar

[7]

H. Brezis, Inéquations d'évolution abstraites, C. R. Acad. Sci. Paris Sér. A-B, 264 (1967), A732-A735.  Google Scholar

[8]

C. Byrne, Y. Censor, A. Gibali and S. Reich, The split common null point problem, J. Nonlinear Convex Anal., 13 (2012), 759-775.   Google Scholar

[9]

L.-C. Ceng and S. Huang, Existence theorems for generalized vector variational inequalities with a variable ordering relation, J. Global Optim., 46 (2010), 521-535.  doi: 10.1007/s10898-009-9436-9.  Google Scholar

[10]

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.  Google Scholar

[11]

Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017.  Google Scholar

[12]

Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 59 (2012), 301-323.  doi: 10.1007/s11075-011-9490-5.  Google Scholar

[13]

C. Ciarcià and P. Daniele, New existence theorems for quasi-variational inequalities and applications to financial models, European J. Oper. Res., 251 (2016), 288-299.  doi: 10.1016/j.ejor.2015.11.013.  Google Scholar

[14]

M. Chen and C. Huang, A power penalty method for a class of linearly constrained variational inequality, J. Ind. Manag. Optim., 14 (2018), 1381-1396.  doi: 10.3934/jimo.2018012.  Google Scholar

[15]

M. G. Cojocaru, P. Daniele and A. Nagurney, Projected dynamical systems and evolutionary variational inequalities via Hilbert spaces with applications, J. Optim. Theory Appl., 127 (2005), 549-563.  doi: 10.1007/s10957-005-7502-0.  Google Scholar

[16]

M.-G. Cojocaru and L. B. Jonker, Existence of solutions to projected differential equations in Hilbert spaces, Proc. Amer. Math. Soc., 132 (2004), 183-193.  doi: 10.1090/S0002-9939-03-07015-1.  Google Scholar

[17]

S. Dafermos, Traffic equilibrium and variational inequalities, Transportation Sci., 14 (1980), 42-54.  doi: 10.1287/trsc.14.1.42.  Google Scholar

[18]

P. Daniele, Time-dependent spatial price equilibrium problem: Existence and stability results for the quantity formulation model, J. Global Optim., 28 (2004), 283-295.  doi: 10.1023/B:JOGO.0000026449.29735.3c.  Google Scholar

[19]

P. Daniele, A. Maugeri and W. Oettli, Time-dependent traffic equilibria, J. Optim. Theory Appl., 103 (1999), 543-555.  doi: 10.1023/A:1021779823196.  Google Scholar

[20]

P. Dupuis and H. Ishii, On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications, Stochastics Stochastics Rep., 35 (1991), 31-62.  doi: 10.1080/17442509108833688.  Google Scholar

[21]

P. Dupuis and A. Nagurney, Dynamical systems and variational inequalities, Ann. Oper. Res., 44 (1993), 9-42.  doi: 10.1007/BF02073589.  Google Scholar

[22]

K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann., 266 (1984), 519-537.  doi: 10.1007/BF01458545.  Google Scholar

[23]

G. Fichera, Problemi elastostatici con vincoli unilaterali: Il problema di Signorini con ambique condizioni al contorno, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia (8), 7 (1963/64), 91-140.   Google Scholar

[24]

G. Fichera, Sul problema elastostatico di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), 34 (1963), 138-142.   Google Scholar

[25]

S. Giuffrè, G. Idone and S. Pia, Some classes of projected dynamical systems in Banach spaces and variational inequalities, J. Global Optim., 40 (2008), 119-128.  doi: 10.1007/s10898-007-9173-x.  Google Scholar

[26]

S. Lawphongpanich and D. W. Hearn, Simplical decomposition of the asymmetric traffic assignment problem, Transportation Res. Part B, 18 (1984), 123-133.  doi: 10.1016/0191-2615(84)90026-2.  Google Scholar

[27]

J.-L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493-519.  doi: 10.1002/cpa.3160200302.  Google Scholar

[28]

S.-Y. Matsushita and L. Xu, On finite convergence of iterative methods for variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 161 (2014), 701-715.  doi: 10.1007/s10957-013-0460-z.  Google Scholar

[29]

A. Moudafi, The split common fixed-point problem for demicontractive mappings, Inverse Problems, 26 (2010), 6pp. doi: 10.1088/0266-5611/26/5/055007.  Google Scholar

[30]

A. Nagurney, D. Parkes and P. Daniele, The Internet, evolutionary variational inequalities, and the time-dependent Braess paradox, Comput. Manag. Sci., 4 (2007), 355-375.  doi: 10.1007/s10287-006-0027-7.  Google Scholar

[31]

B. Panicucci, M. Pappalardo and M. Passacantando, A path-based double projection method for solving the asymmetric traffic network equilibrium problem, Optim. Lett., 1 (2007), 171-185.  doi: 10.1007/s11590-006-0002-9.  Google Scholar

[32]

F. Raciti, Equilibrium conditions and vector variational inequalities: A complex relation, J. Global Optim., 40 (2008), 353-360.  doi: 10.1007/s10898-007-9202-9.  Google Scholar

[33]

L. Scrimali and C. Mirabella, Cooperation in pollution control problems via evolutionary variational inequalities, J. Global Optim., 70 (2018), 455-476.  doi: 10.1007/s10898-017-0580-3.  Google Scholar

[34]

Y. Shehu and O. Iyiola, On a modified extragradient method for variational inequality problem with application to industrial electricity production, J. Ind. Manag. Optim., 15 (2019), 319-342.  doi: 10.3934/jimo.2018045.  Google Scholar

[35]

M. J. Smith, The existence, uniqueness and stability of traffic equilibria, Transportation Res., 13 (1979), 295-304.  doi: 10.1016/0191-2615(79)90022-5.  Google Scholar

[36]

G. Stampacchia, Formes bilinéaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris, 258 (1964), 4413-4416.   Google Scholar

[37]

X. K. Sun and S. J. Li, Duality and gap function for generalized multivalued $\epsilon$-vector variational inequality, Appl. Anal., 92 (2013), 482-492.  doi: 10.1080/00036811.2011.628940.  Google Scholar

[38]

J. Yang, Dynamic power price problem: An inverse variational inequality approach, J. Ind. Manag. Optim., 4 (2008), 673-684.  doi: 10.3934/jimo.2008.4.673.  Google Scholar

Figure 1.  Transportation network patterns of the City $ X\; \text{and}\; Y $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Traffic network pattern of two cities $ X $ and $ Y $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Traffic network pattern of City $ Y $
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Numerical Results
$ t_{i} $ $ x_{1}^{*}(t_{i}) $ $ x_{2}^{*}(t_{i}) $ $ x_{3}^{*}(t_{i}) $ $ x_{4}^{*}(t_{i}) $
$ 0 $ 1 1 1.5 1.5
$ \frac{1}{8} $ 1.125 1.125 1.6875 1.6875
$ \frac{1}{4} $ 1.25 1.25 1.875 1.875
$ \frac{3}{8} $ 1.375 1.375 2.0625 2.0625
$ \frac{1}{2} $ 1.5 1.5 2.25 2.25
$ \frac{5}{8} $ 1.625 1.625 2.4375 2.4375
$ \frac{3}{4} $ 1.75 1.75 2.625 2.625
$ \frac{7}{8} $ 1.875 1.875 2.8125 2.8125
$ 1 $ 2 2 3 3
$ \frac{9}{8} $ 2.125 2.125 3.1875 3.1875
$ \frac{5}{4} $ 2.25 2.25 3.375 3.375
$ \frac{11}{8} $ 2.375 2.375 3.5625 3.5625
$ \frac{3}{2} $ 2.5 2.5 3.75 3.75
$ \frac{13}{8} $ 2.625 2.625 3.9375 3.9375
$ \frac{7}{4} $ 2.75 2.75 4.125 4.125
$ \frac{15}{8} $ 2.875 2.875 4.3125 4.3125
$ 2 $ 3 3 4.5 4.5
Table Options

Download as excel

$ t_{i} $ $ x_{1}^{*}(t_{i}) $ $ x_{2}^{*}(t_{i}) $ $ x_{3}^{*}(t_{i}) $ $ x_{4}^{*}(t_{i}) $
$ 0 $ 1 1 1.5 1.5
$ \frac{1}{8} $ 1.125 1.125 1.6875 1.6875
$ \frac{1}{4} $ 1.25 1.25 1.875 1.875
$ \frac{3}{8} $ 1.375 1.375 2.0625 2.0625
$ \frac{1}{2} $ 1.5 1.5 2.25 2.25
$ \frac{5}{8} $ 1.625 1.625 2.4375 2.4375
$ \frac{3}{4} $ 1.75 1.75 2.625 2.625
$ \frac{7}{8} $ 1.875 1.875 2.8125 2.8125
$ 1 $ 2 2 3 3
$ \frac{9}{8} $ 2.125 2.125 3.1875 3.1875
$ \frac{5}{4} $ 2.25 2.25 3.375 3.375
$ \frac{11}{8} $ 2.375 2.375 3.5625 3.5625
$ \frac{3}{2} $ 2.5 2.5 3.75 3.75
$ \frac{13}{8} $ 2.625 2.625 3.9375 3.9375
$ \frac{7}{4} $ 2.75 2.75 4.125 4.125
$ \frac{15}{8} $ 2.875 2.875 4.3125 4.3125
$ 2 $ 3 3 4.5 4.5
Table 2.  Numerical Results
$ t_{i} $ $ y_{1}^{*}(t_{i}) $ $ y_{2}^{*}(t_{i}) $ $ y_{3}^{*}(t_{i}) $ $ y_{4}^{*}(t_{i}) $ $ y_{5}^{*}(t_{i}) $
$ 0 $ 2.5 2.5 2 2 2
$ \frac{1}{8} $ 2.8125 2.8125 2.25 2.25 2.25
$ \frac{1}{4} $ 3.125 3.125 2.5 2.5 2.5
$ \frac{3}{8} $ 3.4375 3.4375 2.75 2.75 2.75
$ \frac{1}{2} $ 3.75 3.75 3 3 3
$ \frac{5}{8} $ 4.0625 4.0625 3.25 3.25 3.25
$ \frac{3}{4} $ 4.375 4.375 3.5 3.5 3.5
$ \frac{7}{8} $ 4.6875 4.6875 3.75 3.75 3.75
$ 1 $ 5 5 4 4 4
$ \frac{9}{8} $ 5.3125 5.3125 4.25 4.25 4.25
$ \frac{5}{4} $ 5.625 5.625 4.5 4.5 4.5
$ \frac{11}{8} $ 5.9375 5.9375 4.75 4.75 4.75
$ \frac{3}{2} $ 6.25 6.25 5 5 5
$ \frac{13}{8} $ 6.5625 6.5625 5.25 5.25 5.25
$ \frac{7}{4} $ 6.875 6.875 5.5 5.5 5.5
$ \frac{15}{8} $ 7.1875 7.1875 5.75 5.75 5.75
$ 2 $ 7.5 7.5 6 6 6
Table Options

Download as excel

$ t_{i} $ $ y_{1}^{*}(t_{i}) $ $ y_{2}^{*}(t_{i}) $ $ y_{3}^{*}(t_{i}) $ $ y_{4}^{*}(t_{i}) $ $ y_{5}^{*}(t_{i}) $
$ 0 $ 2.5 2.5 2 2 2
$ \frac{1}{8} $ 2.8125 2.8125 2.25 2.25 2.25
$ \frac{1}{4} $ 3.125 3.125 2.5 2.5 2.5
$ \frac{3}{8} $ 3.4375 3.4375 2.75 2.75 2.75
$ \frac{1}{2} $ 3.75 3.75 3 3 3
$ \frac{5}{8} $ 4.0625 4.0625 3.25 3.25 3.25
$ \frac{3}{4} $ 4.375 4.375 3.5 3.5 3.5
$ \frac{7}{8} $ 4.6875 4.6875 3.75 3.75 3.75
$ 1 $ 5 5 4 4 4
$ \frac{9}{8} $ 5.3125 5.3125 4.25 4.25 4.25
$ \frac{5}{4} $ 5.625 5.625 4.5 4.5 4.5
$ \frac{11}{8} $ 5.9375 5.9375 4.75 4.75 4.75
$ \frac{3}{2} $ 6.25 6.25 5 5 5
$ \frac{13}{8} $ 6.5625 6.5625 5.25 5.25 5.25
$ \frac{7}{4} $ 6.875 6.875 5.5 5.5 5.5
$ \frac{15}{8} $ 7.1875 7.1875 5.75 5.75 5.75
$ 2 $ 7.5 7.5 6 6 6
[1]

Dang Van Hieu. Projection methods for solving split equilibrium problems. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2331-2349. doi: 10.3934/jimo.2019056
[2]

Chibueze Christian Okeke, Abdulmalik Usman Bello, Lateef Olakunle Jolaoso, Kingsley Chimuanya Ukandu. Inertial method for split null point problems with pseudomonotone variational inequality problems. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021037
[3]

Annamaria Barbagallo, Rosalba Di Vincenzo, Stéphane Pia. On strong Lagrange duality for weighted traffic equilibrium problem. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1097-1113. doi: 10.3934/dcds.2011.31.1097
[4]

Kenji Kimura, Yeong-Cheng Liou, David S. Shyu, Jen-Chih Yao. Simultaneous system of vector equilibrium problems. Journal of Industrial & Management Optimization, 2009, 5 (1) : 161-174. doi: 10.3934/jimo.2009.5.161
[5]

Jean-Baptiste Burie, Ramsès Djidjou-Demasse, Arnaud Ducrot. Slow convergence to equilibrium for an evolutionary epidemiology integro-differential system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2223-2243. doi: 10.3934/dcdsb.2019225
[6]

Zaki Chbani, Hassan Riahi. Existence and asymptotic behaviour for solutions of dynamical equilibrium systems. Evolution Equations & Control Theory, 2014, 3 (1) : 1-14. doi: 10.3934/eect.2014.3.1
[7]

Xiaona Fan, Li Jiang, Mengsi Li. Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1795-1807. doi: 10.3934/jimo.2018123
[8]

O. Chadli, Z. Chbani, H. Riahi. Recession methods for equilibrium problems and applications to variational and hemivariational inequalities. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 185-196. doi: 10.3934/dcds.1999.5.185
[9]

Grace Nnennaya Ogwo, Chinedu Izuchukwu, Oluwatosin Temitope Mewomo. A modified extragradient algorithm for a certain class of split pseudo-monotone variational inequality problem. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021011
[10]

Zeng-Zhen Tan, Rong Hu, Ming Zhu, Ya-Ping Fang. A dynamical system method for solving the split convex feasibility problem. Journal of Industrial & Management Optimization, 2021, 17 (6) : 2989-3011. doi: 10.3934/jimo.2020104
[11]

Francis Akutsah, Akindele Adebayo Mebawondu, Hammed Anuoluwapo Abass, Ojen Kumar Narain. A self adaptive method for solving a class of bilevel variational inequalities with split variational inequality and composed fixed point problem constraints in Hilbert spaces. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021046
[12]

Hongyu He, Naohiro Kato. Equilibrium submanifold for a biological system. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1429-1441. doi: 10.3934/dcdss.2011.4.1429
[13]

J. Gwinner. On differential variational inequalities and projected dynamical systems - equivalence and a stability result. Conference Publications, 2007, 2007 (Special) : 467-476. doi: 10.3934/proc.2007.2007.467
[14]

Yu Han, Nan-Jing Huang. Some characterizations of the approximate solutions to generalized vector equilibrium problems. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1135-1151. doi: 10.3934/jimo.2016.12.1135
[15]

Pablo Amster, Mariel Paula Kuna, Gonzalo Robledo. Multiple solutions for periodic perturbations of a delayed autonomous system near an equilibrium. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1695-1709. doi: 10.3934/cpaa.2019080
[16]

Kei Fong Lam, Hao Wu. Convergence to equilibrium for a bulk–surface Allen–Cahn system coupled through a nonlinear Robin boundary condition. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1847-1878. doi: 10.3934/dcds.2020096
[17]

Elvio Accinelli, Bruno Bazzano, Franco Robledo, Pablo Romero. Nash Equilibrium in evolutionary competitive models of firms and workers under external regulation. Journal of Dynamics & Games, 2015, 2 (1) : 1-32. doi: 10.3934/jdg.2015.2.1
[18]

Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285
[19]

Peiyu Li. Solving normalized stationary points of a class of equilibrium problem with equilibrium constraints. Journal of Industrial & Management Optimization, 2018, 14 (2) : 637-646. doi: 10.3934/jimo.2017065
[20]

Gang Qian, Deren Han, Lingling Xu, Hai Yang. Solving nonadditive traffic assignment problems: A self-adaptive projection-auxiliary problem method for variational inequalities. Journal of Industrial & Management Optimization, 2013, 9 (1) : 255-274. doi: 10.3934/jimo.2013.9.255

2020 Impact Factor: 1.801

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy