We propose a vector optimization approach to linear Cournot oligopolistic market equilibrium models where the strategy sets depend on each other. We use scalarization technique to find a Pareto efficient solution to the model by using a jointly constrained bilinear programming formulation. We then propose a decomposition branch-and-bound algorithm for globally solving the resulting bilinear problem. The subdivision takes place in one-dimensional intervals that enables solving the problem with relatively large sizes. Numerical experiments and results on randomly generated data show the efficiency of the proposed algorithm.
Mots clés : Generalized Cournot model, bilinear programming, branch-and-bound, Pareto solution
@article{RO_2015__49_5_845_0,
author = {Van Quy, Nguyen},
title = {A jointly constrained bilinear programming method for solving generalized {Cournot{\textendash}Pareto} models},
journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
pages = {845--864},
publisher = {EDP-Sciences},
volume = {49},
number = {5},
year = {2015},
doi = {10.1051/ro/2015031},
mrnumber = {3549616},
zbl = {1338.91090},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ro/2015031/}
}
TY - JOUR AU - Van Quy, Nguyen TI - A jointly constrained bilinear programming method for solving generalized Cournot–Pareto models JO - RAIRO - Operations Research - Recherche Opérationnelle PY - 2015 SP - 845 EP - 864 VL - 49 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ro/2015031/ DO - 10.1051/ro/2015031 LA - en ID - RO_2015__49_5_845_0 ER -
%0 Journal Article %A Van Quy, Nguyen %T A jointly constrained bilinear programming method for solving generalized Cournot–Pareto models %J RAIRO - Operations Research - Recherche Opérationnelle %D 2015 %P 845-864 %V 49 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ro/2015031/ %R 10.1051/ro/2015031 %G en %F RO_2015__49_5_845_0
Van Quy, Nguyen. A jointly constrained bilinear programming method for solving generalized Cournot–Pareto models. RAIRO - Operations Research - Recherche Opérationnelle, Tome 49 (2015) no. 5, pp. 845-864. doi : 10.1051/ro/2015031. http://www.numdam.org/articles/10.1051/ro/2015031/
, Jointly constrained bilinear programs and related problems: An overview. Comput. Math. Appl. 19 (1990) 53–62. | DOI | MR | Zbl
, and , Gap functions for quasivariational inequalities and generalized Nash equilibrium problems. J. Optim. Theory Optim. 151 (2011) 474–488. | DOI | MR | Zbl
, , and , Existence and solution methods for equilibria. Eur. J. Oper. Res. 227 (2013) 1–11. | DOI | MR | Zbl
and , From optimization and variational inequality to equilibrium problems. Princeton University Press, 1963. Math. Stud. 63 (1994) 127–149. | MR | Zbl
F. Facchinei and J.S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2003). | MR | Zbl
and , Generalized Nash equilibrium problems. Ann. Oper. Res. 175 (2010) 177–211. | DOI | MR | Zbl
, and , Solving quasi-variational inequalities via their KKT conditions. Math. Program. 144 Ser. A (2014) 369–412. | DOI | MR | Zbl
, A class of gap functions for quasi-variational inequlity problems. J. Ind. Manag. Optim. 3 (2007) 165–174. | DOI | MR | Zbl
and , Quasi-variational inequality, generalized Nash equilibria, and multi-leader-folower games. Comput. Manag. Sci. 2 (2005) 21–26. | DOI | MR | Zbl
and , Gap functions and error bounds for quasi-variational inequalities. J. Global Optim. 53 (2012) 737–748. | DOI | MR | Zbl
, and , On a smooth dual gap function for a class of quasi-variational inequalities. J. Optim. Theory Appl. 163 (2014) 413–438. | DOI | MR | Zbl
, and , Smoothless properties of a regularized gap function for quasi-variational inequalities. Optim. Methods Softw 29 (2014) 720–750. | DOI | MR | Zbl
R. Horst and H. Tuy, Global Optimization, Deterministic Approach. Springer, Berlin (1990). | MR | Zbl
I. Konnov, Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2001). | MR | Zbl
and , Gap function approach to the generalized Nash equilibrium problem. J. Optim. Theory Appl. 144 (2010) 511–531. | DOI | MR | Zbl
, and , A mathematical programming approach for determining oligopolistic market equilibrium. Math. Program. 24 (1982) 92–106. | DOI | MR | Zbl
and , A method for minimizing a convex-concave function over a convex set. J. Optim. Theory Appl. 70 (1990) 377–384. | DOI | MR | Zbl
and , A global optimization method for solving convex quadratic bilevel programming problems. J. Global Optim. 26 (2003) 199–219. | DOI | MR | Zbl
, and , On Nash−Cournot oligopolistic market models with concave cost functions. J. Global Optim. 41 (2007) 351–364 | DOI | MR | Zbl
D.T. Luc, Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, New York Berlin, Heidelberg (1989). | MR | Zbl
, and , Scalarizing Functions for Generating the Weakly Efficient Solution Set in Convex Multiobjective Problems. SIAM J. Optim. 15 (2005) 987–1001. | DOI | MR | Zbl
A. Nagurney, Network Economics: A Variational Inequality Approach. Kluwer Academic Publishers (1993). | MR
and , Note on non-cooperative convex games. Pacific J. Math. 5 (1955) 807–815. | DOI | MR | Zbl
and , A splitting proximal point method for Nash–Cournot equilibrium models involving nonconvex cost functions. J. Nonlin. Convex Anal. 12 (2011) 519–534. | MR | Zbl
, A vector optimization approach to Cournot oligopolistic market Models. Int. J. Optim.: Theory, Methods Appl. 1 (2009) 341–360. | MR | Zbl
, and , A new class of hybrid extragradient algorithms for solving quasi- equilibrium problems. J. Global Optim. 56 (2013) 373–397. | DOI | MR | Zbl
P.L. Yu, Multiple − Criteria Decision Making: Concepts, Techniques and Extensions. Plenum Press, New York, London (1985). | MR | Zbl
Cité par Sources :
