Abstract
We use techniques from global optimization to develop an algorithm for finding a global solution of nonconvex mixed variational inequality problems involving separable DC cost functions. In contrast to the convex mixed variational inequality, in these problems, a local solution may not be a global one. The proposed algorithm uses the convex envelope of the separable cost function over boxes to approximate a DC cost problem with a convex cost one that can be solved by available methods. To obtain better approximate solutions, the algorithm uses an adaptive rectangular bisection which is performed only in the space of concave variables. The algorithm is applied to solve the Nash-Cournot and Bertrand equilibrium models with logarithm and quadratic concave costs. Computational results on a lot number of randomly generated data show that the proposed algorithm is efficient for these models, when the number of the concave cost functions is moderate, while the size of the model may be much larger.
Similar content being viewed by others
References
Aussel, D., Correa, R., Marechal, M.: Gap functions for quasivariational inequalities and generalized Nash equilibrium problems. J. Optim. Theory Appl. 151, 474–489 (2011)
Bigi, G., Passacantando, M.: Diferentiated oligopolistic markets with concave cost functions via Ky Fan inequalities. Decis. Econ. Finance 40, 63–79 (2017)
Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Nonlinear Programming Techniques for Equilibria. Springer (2018)
Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2002)
Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. Ann. Oper. Res. 175, 177–211 (2010)
Fukushima, M.: A class of gap functions for quasi-variational inequlity problems. J. Ind. Manag. Optim. 3, 165–174 (2007)
Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Prog. 53, 99–110 (1992)
Fukushima, M., Pang, J.S.: Quasi-variational inequality, generalized Nash equilibria, and multi-leader-folower games. Comput. Manag. Science 2, 21–26 (2005)
Hillestad, R.J., Jacobsen, S.E.: Linear programswith an additional reverse convex constraint. Appl. Math. Optim. 6, 257–269 (1980)
Horst, R., Tuy, H.: Global Optimization (Deterministic Approach). Springer, Berlin (1990)
Konno, H.: A cutting plane for solving bilinear progrmming. Math. Prog. 11, 14–27 (1976)
Konnov, I.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2001)
Kubota, K., Fukushima, M.: Gap function approach to the generalized Nash equilibrium problem. J. Optim. Theory Appl. 144, 511–531 (2010)
Murphy, H.F., Sherali, H.D., Soyster, A.L.: A mathematical programming approach for determining oligopolistic market equilibrium. Math. Prog. 24, 92–106 (1982)
Muu, L.D., Oettli, W.: A method for minimizing a convex-concave function over a convex set. J. Optim. Theory Appl. 70, 377–384 (1990)
Muu, L.D.: An algorithm for solving convex programs with an additional convex-concave constraint. Math. Prog. 61, 75–87 (1993)
Muu, L.D.: Convex-concave progrmming as a decomposition approach to global optimization. Acta Math. Vietnam. 18, 61–77 (1993)
Muu, L.D., Nguyen, V.H., Quy, N.V.: On Nash-Cournot oligopolistic market models with concave cost functions. J. Glob. Optim. 41, 351–364 (2007)
Muu, L.D., Quoc, T.D.: One step from DC optimization to DC mixed variational inequalities. Optimization 59, 63–76 (2010)
Muu, L.D., Quoc, T.D.: Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash-Cournot Equilibrium. Model. J. Optim. Theory Appl. 142, 185–204 (2009)
Nagurney, A.: Network Economics: A Variational Inequality Approach. Kluwer Academic Publishers (1993)
Pham, D.T., Le Thi, H.A.: Convex analysis approach to DC prohramming: theory, algorithms and applications. Acta Math. Vietnam. 22, 289–355 (1997)
Pardalos, P.M., Rosen, J.B.: Constrained Global Optimization: Algorithms and Applications Lecture Notes in Computer Sciences, vol. 268. Springer, Berlin (1987)
Quoc, T.D., Muu, L.D.: A spritting proximal point method for Nash-Cournot equilibrium models involving nonconvex cost functions. J. Nonlinear Convex Anal. 12, 519–534 (2011)
Quy, N.V.: A vector optimization approach to Cournot oligopolistic market models. Inter. J. Optim.: Theory, Meth. Appl. 1, 341–360 (2009)
Rockafellar, R.T.: Convex Analysis. Princeton University Press (1970)
Sun, W.-Y., Sampaio, R.J.B., Condido, M.A.B.: Proximal point algorithm for minimization of DC function. J. Comput. Math. 21, 451–462 (2003)
Tuy, H.: Concave programming under linear constraints. Sov. Math. Dokl. 5, 1437–1440 (1964)
Tuy, H.: Global minimization of difference of two convex functions. In: Lecturee Note in Economics and Math. Systems, vol. 226, pp 98–108. Springer, Berlin (1984)
Tuy, H.: Monotonic optimization: problems and solution approaches. SIAM J. Optim. 11, 464–494 (2000)
Tuy, H.: Convex Analysis and Global Optimization, 2nd edition. Springer (2016)
Acknowledgements
We would like to thank the editor and referees for their valuable comments, suggestions, and remarks that helped us very much to improve the quality of the paper.
Funding
The first author of this paper appreciated the support from the NAFOSTED, under grant 101.01-2017.315.
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper is dedicated to Professor Hoang Tuy on the occasion of his 90th birthday
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Muu, L.D., Van Quy, N. Global Optimization from Concave Minimization to Concave Mixed Variational Inequality. Acta Math Vietnam 45, 449–462 (2020). https://doi.org/10.1007/s40306-020-00363-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40306-020-00363-5
Keywords
- DC optimization
- Nonconvex mixed variational inequality
- Nash-Cournot oligopolistic model
- Concave cost
- Global solution
- Gap function
- Convex envelope