# zbMATH — the first resource for mathematics

On the convergence of descent methods for monotone variational inequalities. (English) Zbl 0828.90127
Recently, Zhu and Marcotte (1993) established the convergence of a modified descent algorithm for monotone variational inequalities. Using algorithmic equivalence results due to Patriksson (1993) and Larsson and Patriksson (1994), we show that this convergence result may be used to establish the convergence of slightly modified versions of the classical successive approximation algorithms of Dafermos (1983) and Cohen (1988), and of the descent algorithms of Wu, Florian and Marcotte (1993), Patriksson (1993), and Larsson and Patriksson (1994), under assumptions that are both much milder and much easier to verify than those for their original statements.

##### MSC:
 90C30 Nonlinear programming 49J40 Variational inequalities
Full Text:
##### References:
 [1] Cohen, G., Auxiliary problem principle extended to variational inequalities, J. optim. theory appl., 59, 325-333, (1988) · Zbl 0628.90066 [2] Dafermos, S., An iterative scheme for variational inequalities, Math. programming, 26, 40-47, (1983) · Zbl 0506.65026 [3] Fukushima, M., Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Math. programming, 53, 99-110, (1992) · Zbl 0756.90081 [4] Harker, P.T.; Pang, J.-S., Finite-dimensional variational inequality and nonlinear complementary problems: a survey of theory, algorithms and applications, Math. programming, 48, 161-220, (1990) · Zbl 0734.90098 [5] Heydecker, B.G., Some consequences of detailed junction modeling in road traffic assignment, Transportation sci., 17, 263-281, (1983) [6] Korpelevich, G.M., The extragradient method for finding saddle points and other problems, Matekon, 13, 35-49, (1977) [7] Larsson, T.; Patriksson, M., A class of gap functions for variational inequalities, Math. programming, 64, 53-79, (1994) · Zbl 0819.65101 [8] Marcotte, P.; Dussault, J.-P., A modified Newton method for solving variational inequalities, (), 1433-1436 [9] Pang, J.-S.; Chan, D., Iterative methods for variational and complementarity problems, Math. programming, 24, 284-313, (1982) · Zbl 0499.90074 [10] Patriksson, M., A unified description of iterative algorithms for traffic equilibria, Eur. J. oper. res., 71, 154-176, (1993) · Zbl 0802.90074 [11] Patriksson, M., A unified framework of descent algorithms for nonlinear programs and variational inequalities, () · Zbl 0904.49007 [12] Patriksson, M., The traffic assignment problem: models and methods, (1994), VSP Utrecht, The Netherlands · Zbl 0828.90127 [13] Smith, M.J., A descent algorithm for solving monotone variational inequalities and monotone complementarity problems, J. optim. theory appl., 44, 485-496, (1984) · Zbl 0535.49023 [14] Wu, J.H.; Florian, M.; Marcotte, P., A general descent framework for the monotone variational inequality problem, Math. programming, 61, 281-300, (1993) · Zbl 0813.90111 [15] Zhu, D.L.; Marcotte, P., Modified descent methods for solving the monotone variational inequality problem, Oper. res. lett., 14, 111-120, (1993) · Zbl 0795.49011 [16] Zuhovickiǐ, S.I.; Polyak, R.A.; Primak, M.E., Two methods of search for equilibrium points of n-person concave games, Soviet math. doklady, 10, 279-282, (1969) · Zbl 0191.49801
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.