zbMATH — the first resource for mathematics

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.

58J35 Heat and other parabolic equation methods for PDEs on manifolds
49J40 Variational inequalities
Full Text: DOI
[1] Apostol, T.M., Mathematical analysis, (1974), Addison-Wesley Publishing Company · Zbl 0126.28202
[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, 2, 118-124, (1989)
[4] Eaves, B.C., On the basic theorem of complementarity, Mathematical programming, 1, 68-75, (1971) · Zbl 0227.90044
[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
[6] Facchinei, F.; Pang, J.S., Finite-dimensional variational inequalities and complementarity problems, Springer series in operations research, (2003), Springer-Verlag · Zbl 1062.90002
[7] Fletcher, R., Practical methods of optimization, (1987), John Wiley & Sons · Zbl 0905.65002
[8] Goldstein, A.A., Convex programming in Hilbert space, American mathematical society bulletin, 70, 709-710, (1964) · Zbl 0142.17101
[9] He, B.S., Inexact implicit methods for monotone general variational inequalities, Mathematical programming, 86, 199-217, (1999) · Zbl 0979.49006
[10] He, B.S., Solving a class of linear projection equations, Numerische Mathematik, 68, 1, 71-80, (1994) · Zbl 0822.65040
[11] He, B.S.; Liao, L.-Z., Improvements of some projection methods for monotone nonlinear variational inequalities, Journal of optimization theory and applications, 112, 1, 111-128, (2002) · Zbl 1025.65036
[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, 1, 129-143, (2002) · Zbl 0998.65066
[13] Levitin, E.S.; Polyak, B.T., Constrained minimization problems, USSR computational mathematics and mathematical physics, 6, 1-50, (1966) · Zbl 0161.07002
[14] Marcotte, P.; Margquis, G.; Zubieta, L., A Newton-SOR method for spatial price equilibrium, Transportation science, 26, 1, 36-47, (1992) · Zbl 0762.90016
[15] Martinet, B., Regularization d’inequations variationelles par approximations sucessives, Revue francaise d’informatique et de recherche opérationelle, 4, 154-159, (1970)
[16] Nagurney, A., An algorithm for the single commodity spatial price equilibrium problem, Regional science and urban economics, 16, 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
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.