A proximal point algorithm for solving a class of implicit equilibrium models
Main Article Content
Abstract
We apply the proximal point algorithm to solve a class of implicit price equilibrium models including the Walras supply-demand and competitive equilibrium ones, where both supply and demand are given
implicitly as the solution-sets of mathematical programs depending on the price. Such models are formulated as complementarity or variational inequality forms. We employ a monotonicity property of the cost operator to develop proximal point based algorithms to approximate an equilibrium point of the model. Convergence of the algorithm is proved and some computational results with many randomly generated data are reported to show that the proposed algorithms work well for this class of equilibrium models.
Article Details
Keywords
Walras, competitive models, implicit supply-demand, proximal point based algorithm, complementarity, variational inequality
References
[2] P. N. Anh, H. A. Le Thi (2013) An Armijo-type method for pseudomonotone 1equilibrium problems and its applications, JGlob. Optim.57:803820 DOI 10.1007/s10898-012-9970-8
[3] P. N. Anh (2024), Strong Quasi-nonexpansiveness of solution mappings ofequilibrium problems, Vietnam J. Math. https://doi.org/10.1007/s10013-024-00697-9.
[4] H. H. Bauschke, P.L.Combettes (2011) Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York.
[5] C. Berge (1968) Topological Spaces, MacMillan, New York.
[6] G. Bigi, , M. Castellani, M. Pappalardo, M. Passacantando (2019) Nonlinear Programming Techniques for Equilibria, Springer.
[7] F. Deutsch, I. Yamada(1998) Minimizing certain convex functions overthe intersection of the fixed point sets of nonexpansive mappings, Numer. Funct. Anal, Optim.19,33–56. A proximal point algorithm ... 13
[8] B. V. Dinh, P. G. Hung and L. D. Muu (2014) Bilevel optimization as a regularization approach to pseudomonotone equilibrium problems, Numer. Funct. Anal. Optim. 35, 539-563.
[9] P. M. Duc, L. D. Muu (2016) A splitting algorithm for a class of bilevel equilibrium problems involving nonexpansive mappings, Optimization 65, 1855-1866.
[10] F. Facchinei, J.-S. Pang (2003): Finite-Dimensional VariationalInequalities and Complementarity Problems, Vol. 1,2, Springer-Verlag, Berlin.
[11] E.G. Goldstein, N. V. Tretyakov (1996), Modified Lagrangians and Monotone Maps in Optimization, Wiley, New York, NY.
[12] N.N. Hai, L.D. Muu, B. V. Dinh (2023), An algorithm for quasiconvex equilibrium problems and asymptotically nonexpansive mappings: application to a Walras model with implicit supply–demand, Math. Meth. Oper. Res.,https://doi.org/10.1007/s00186-023-00837-w
[13] I. V. Konnov (2000) Combined Relaxation Methods for Variational Inequalities, Springer-Verlag, Berlin.
[14] J. K. Kim, P.N Anh, H.g. Hyun (2012) A proximal point-type algorithm for pseudomonotone equilibrium problem, Bull. Korean Math. Soc.49, 749-759.
[15] I. V. Konnov (2007): Economics Models and Variational Inequalities, ELSEVIER
[16] L. D. Muu and T. D. Quoc (2009), Regularization algorithms for solvingmonotone Ky Fan inequalities with application to a Nash-Cournotequilibrium model. Optim. Theory Appl.,142, 185-204.
[17] Quoc, T.D., Muu, L.D., Nguyen, V.H. (2008): Extragradient algorithms extended to equilibrium problems. Optim., 57, 749-776.
[18] R. T. Rockafellar (1970): Convex Analysis. Princeton University Press, Princeton, New Jersey.
[19] R.T. Rockafellar (1976), Monotone operator and the proximal point algorithm, SIAM J. control and Optim. 14, 877-898.
[20] H.E. Scarf and T. Hansen (1973) Computation of Economic Equilibria,
Yale University Press, New Haven.
[21] H. Tuy (2016) Convex Analysis and Global Optimization, Springer,Berlin.
[22] L.Walras (1874) El´ements d’´Economie Politique Pure, L. Corbaz, Lausanne;English translation (1954) Elements of Pure Economics, Allen and Unwin, London.
[23] HK. Xu (2002) Iterative algorithms for nonlinear operators. J. LondonMath. Soc. 66, 240-256.
[24] I. Yamada, N. Ogura (2005) Hybrid steepest descent method forvariational inequality problem over the fixed point set of certain quasinonexpansive mappings, Numer. Funct. Anal. Optim. 25, 178-189.
[25] D. Zhu, P. Marcotte (1995) A new class of generalized monotonicity,
J. Optim. Theory Appl. 67, 457-471.