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Self-adaptive projection method for co-coercive variational inequalities. (English) Zbl 1163.58305
Summary: In some real-world problems, the mapping of the variational inequalities does not have any explicit forms and only the function value can be evaluated or observed for given variables. In this case, if the mapping is co-coercive, the basic projection method is applicable. However, in order to determine the step size, the existing basic projection method needs to know the co-coercive modulus in advance. In practice, usually even if the mapping can be characterized co-coercive, it is difficult to evaluate the modulus, and a conservative estimation will lead an extremely slow convergence. In view of this point, this paper presents a self-adaptive projection method without knowing the co-coercive modulus. We also give a real-life example to demonstrate the practicability of the proposed method.
MSC:
58J35Heat and other parabolic equation methods for PDEs on manifolds
49J40Variational methods including variational inequalities
References:
[1]Apostol, T. M.: Mathematical analysis, (1974) · Zbl 0309.26002
[2]Armijo, L.: Minimization of functions having continuous partial derivatives, Pacific journal of mathematics 16, 1-3 (1966) · Zbl 0202.46105
[3]Dafermos, S.; Nagurney, A.: Supply and demand equilibration algorithms for a class of market equilibrium problems, Transportation science 23, No. 2, 118-124 (1989)
[4]Eaves, B. C.: On the basic theorem of complementarity, Mathematical programming 1, 68-75 (1971) · Zbl 0227.90044 · doi:10.1007/BF01584073
[5]Eckstein, J.; Bertsekas, D. P.: On the Douglas – Rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical programming 55, 293-318 (1992) · Zbl 0765.90073 · doi:10.1007/BF01581204
[6]Facchinei, F.; Pang, J. S.: Finite-dimensional variational inequalities and complementarity problems, (2003)
[7]Fletcher, R.: Practical methods of optimization, (1987) · Zbl 0905.65002
[8]Goldstein, A. A.: Convex programming in Hilbert space, American mathematical society bulletin 70, 709-710 (1964) · Zbl 0142.17101 · doi:10.1090/S0002-9904-1964-11178-2
[9]He, B. S.: Inexact implicit methods for monotone general variational inequalities, Mathematical programming 86, 199-217 (1999) · Zbl 0979.49006 · doi:10.1007/s101070050086
[10]He, B. S.: Solving a class of linear projection equations, Numerische Mathematik 68, No. 1, 71-80 (1994) · Zbl 0822.65040 · doi:10.1007/s002110050048
[11]He, B. S.; Liao, L. -Z.: Improvements of some projection methods for monotone nonlinear variational inequalities, Journal of optimization theory and applications 112, No. 1, 111-128 (2002) · Zbl 1025.65036 · doi:10.1023/A:1013096613105
[12]He, B. S.; Yang, H.; Meng, Q.; Han, D. R.: Modified goldstein – levitin – Polyak projection method for asymmetric strongly monotone variational inequalities, Journal of optimization theory and applications 112, No. 1, 129-143 (2002) · Zbl 0998.65066 · doi:10.1023/A:1013048729944
[13]Levitin, E. S.; Polyak, B. T.: Constrained minimization problems, USSR computational mathematics and mathematical physics 6, 1-50 (1966)
[14]Marcotte, P.; Margquis, G.; Zubieta, L.: A Newton-SOR method for spatial price equilibrium, Transportation science 26, No. 1, 36-47 (1992) · Zbl 0762.90016 · doi:10.1287/trsc.26.1.36
[15]Martinet, B.: Regularization d’inequations variationelles par approximations sucessives, Revue francaise d’informatique et de recherche opérationelle 4, 154-159 (1970) · Zbl 0215.21103
[16]Nagurney, A.: An algorithm for the single commodity spatial price equilibrium problem, Regional science and urban economics 16, No. 4, 573-588 (1986)
[17]Rockafellar, R. T.: Monotone operators and the proximal point algorithm, SIAM journal on control and optimization 14, 877-898 (1976) · Zbl 0358.90053 · doi:10.1137/0314056
[18]Samuelson, P. A.: Spatial price equilibrium and linear programming, American economic review 42, 283-303 (1952)
[19]Zhu, T.; Yu, Z. G.: A simple proof for some important properties of the projection mapping, Mathematical inequalities and applications 7, 453-456 (2004) · Zbl 1086.49007