An algorithm for minimizing a strongly convex function on the equilibrium set of price equilibrium models
Nội dung chính của bài viết
Tóm tắt
We consider a class of equilibrium models including the im plicit Walras supply-demand and competitive models that, in general, is ill posed. We formulate such a model in the form a variational inequality hav ing certain monotonicity property which allows us to describe an algorithm avoiding the ill-posedness by finding the equilibrium point that is nearest to the given guessed or desired equilibrium price for the model. A main difficulty of the problem is that its feasible domain is not given explicitly as in a standard convex programming problem. The proposed algorithm is a combination between the gradient one and the Mann-Krashnoschelskii f ixed point procedure. The obtained computational results with many ran domly generated data show that the proposed algorithm works well for this class of the equilibrium models.
Chi tiết bài viết
Từ khóa
Equilibrium model, variational inequality, bilevel optimization algorithm, regularization
Tài liệu tham khảo
[2] Deutsch, F., Yamada, I. Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansivemappings, Numer. Funct. Anal, Optim. 19, 33-56 (1998).
[3] Dinh, B. V., Hung, P. G., Muu, L. D. Bilevel optimization as a regularization approach to pseudomonotone equilibrium problems, Numer. Funct. Anal. Optim. 35, 539-563 (2014)
[4] Facchinei, F., Pang, J. S. Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol. 1,2, Springer-Verlag, Berlin (2003).
[5] Hai, N. N., Muu, L. D., Dinh, B. V. An algorithm for quasiconvex equilibrium problems and asymptotically nonexpansive mappings: application to a Walras model with implicit supply–demand, Math. Methods Oper. Res. 98, 299-324 (2023).
[6] Golshtein, E. G., Tretyakov, N. V. Modifed Lagrangians and Monotone Maps in Optimization, Wiley, New York, NY (1996).
[7] Konnov, I. V. Combined Relaxation Methods for Variational Inequalities, Springer-Verlag, Berlin (2000).
[8] Konnov, I. V. Economics Models and Variational Inequalities, Elsevier (2007).
[9] 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).
[10] Rockafellar, R. T. Monotone operator and the proximal point algorithm, SIAM J. Control and Optim. 14, 877-898 (1976).
[11] Tuy, H. Convex Analysis and Global Optimization, Springer, Berlin (2016).
[12] Walras, L. El´ements d’´Economie Politique Pure (1874), L. Corbaz, Lausanne; English translation.: Elements of Pure Economics, Allen and Unwin, London (1954).
[13] Xu, H. K. Iterative algorithms for nonlinear operators. J. London Math. Soc. 66 : 240-256 (2002).
[14] Yamada, I., Ogura, N. Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings, Numer. Funct. Anal. Optim. 25 , 178-189 (2005).
[15] Zhu, D., Marcotte, P. A new class of generalized monotonicity, J. Optim. Theory Appl. 67, 457-471 (1995).