## On Nash-Cournot oligopolistic market equilibrium models with concave cost functions.(English)Zbl 1146.91029

In this paper oligopolistic market equilibrium models are considered, where the cost functions are assumed to be piecewise linear concave. It is shown, that the task, to find a global equilibrium strategy, can be formulated as a mixed variational inequality problem over a bounded rectangle. An equilibrium point need not exist. The authors prove using (upper semicontinuous mappings and) Kakutani’s fixed point theorem conditions, such that an equilibrium point exists and give a decomposition algorithm in order to find a global equilibrium point (if there is one). This method proceeds by dividing the strategy feasible rectangle set into subrectangles, on each of them the cost function is affine. They test their algorithm for more than 20 problems and show, that it is efficient when the number of the firms with concave cost functions is not too big.

### MSC:

 91B52 Special types of economic equilibria 49J40 Variational inequalities
Full Text:

### References:

 [1] Anh, P.N., Muu, L.D., Nguyen, V.H., Strodiot, J.J.: On the contraction and nonexpensiveness properties of the marginal mapping in generalized variational inequalities involving cocoercive operators. In: Eberhard, A., Hadjisavvas, N., Luc, D.T.(eds). Generalized Convexity, and Generalized Monotonicity and Applications, pp. 89–111, Chapter 5, Springer (2005) · Zbl 1138.90476 [2] Anh P.N., Muu L.D., Nguyen V.H. and Strodiot J.J. (2005). Using the Banach contraction principle to implement the proximal point method for solving multivalued monotone variational inequalities. J. Optim. Theory Appl. 124: 285–306 · Zbl 1062.49005 · doi:10.1007/s10957-004-0926-0 [3] Aubin J.P. and Ekeland I. (1984). Applied Nonlinear Analysis. Wiley, New York · Zbl 0641.47066 [4] Berge C. (1968). Topological Spaces. MacMillan, New York [5] Blum E. and Oettli W. (1994). From optimization and variational inequality to equilibrium problems. Mathe. Stud. 63: 127–149 · Zbl 0888.49007 [6] Dafermos S. (1990). Exchange price equilibria and variational inequalities. Math. Program. 46: 391–402 · Zbl 0709.90013 · doi:10.1007/BF01585753 [7] Dafermosm S. and Nagurney A. (1997). Oligopolistic and competitive behavior of spatially separated markets. Reg. Sci. Urban Econ. 17: 225–254 [8] Cohen G. (1998). Auxiliary problem principle extended to variational inequalities. J. Optim. Theory Appl. 59: 325–333 · Zbl 0628.90066 [9] Facchinei F. and Pang J.S. (2002). Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin · Zbl 1062.90002 [10] Fukushima M. (1992). Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53: 99–110 · Zbl 0756.90081 · doi:10.1007/BF01585696 [11] Harker P.T. and Pang J.S. (1990). Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Mathe. Program. 48: 161–220 · Zbl 0734.90098 · doi:10.1007/BF01582255 [12] Hue T.T., Strodiot J.J. and Nguyen V.H. (2004). Convergence of the Approximate Auxiliary Problem Method for Solving Generalized Variational Inequalities. J. Optim. Theory Appl. 121: 119–145 · Zbl 1056.49013 · doi:10.1023/B:JOTA.0000026134.57920.e1 [13] Horst R. and Tuy H. (1990). Global Optimization (Deterministic Approach). Springer, Berlin · Zbl 0704.90057 [14] Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press (1980) · Zbl 0457.35001 [15] Konnov I. (2001). Combined Relaxation Methods for Variational Inequalities. Springer, Berlin · Zbl 0982.49009 [16] Konnov I. and Kum S. (2001). Descent methods for mixed variational inequalities in a Hilbert space. Nonlinear Anal. Theory Meth. Appl. 47: 561–572 · Zbl 1042.49514 · doi:10.1016/S0362-546X(01)00201-2 [17] Marcotte P. (1995). A new algorithm for solving variational inequalities. Mathematical Programming 33: 339–351 · doi:10.1007/BF01584381 [18] Muu L.D. (1986). An augmented penalty function method for solving a class of variational inequalities. Soviet. Comput. Math. Phys. 12: 1788–1796 · Zbl 0641.90073 [19] Muu W. and Oettli L.D. (1992). Convergence of an adaptive scheme for finding constraint equilibria. Nonlinear Anal Theory Meth Appl. 18: 1159–1166 · Zbl 0773.90092 · doi:10.1016/0362-546X(92)90159-C [20] Muu L.D. and Quy N.V. (2003). A global optimization method for solving convex quadratic bilevel programming problems. J. Global Optim. 26: 199–219 · Zbl 1053.90104 · doi:10.1023/A:1023047900333 [21] Nagurney A. (1993). Network Economics: A Variational Inequality Approach. Kluwer, Academic Publishers · Zbl 0873.90015 [22] Noor M.A. (2001). Iterative schemes for quasimonotone mixed variational inequalities. Optimization 50: 29–44 · Zbl 0986.49008 · doi:10.1080/02331930108844552 [23] Patriksson M. (1999). Nonlinear Programming and Variational Inequality Problems: A Unified Approach. Kluwer, Dordrecht · Zbl 0913.65058 [24] Patriksson M. (1997). Merit function and descent algorithms for a class of variational inequality problems.. Optimization 41: 37–55 · Zbl 0904.49007 · doi:10.1080/02331939708844324 [25] Salmon G., Nguyen V.H. and Strodiot J.J. (2000). Coupling the auxiliary problem principle and epiconvergence theory to solve general variational inequalities. J. Optim Theory Appl. 104: 629–657 · Zbl 1034.90018 · doi:10.1023/A:1004693710334 [26] Taji T. and Fukushima M. (1996). A new merit function and a successive quadratic programming algorithm for variational inequality problem. SIAM J. Optim. 6: 704–713 · Zbl 0853.49011
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.