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The Walrasian Equilibrium and Centralized Distributed Optimization in Terms of Modern Convex Optimization Methods by an Example of the Resource Allocation Problem

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Abstract

The resource allocation problem and its numerical solution are considered. The following is demonstrated: (1) Walrasian price-adjustment mechanism for determining the equilibrium state; (2) decentralized role of prices; (3) Slater’s method for price restrictions (dual Lagrange multipliers); (4) new mechanism for determining equilibrium prices, in which prices are fully controlled by economic agents—nodes (enterprises)—rather than by the Center (Government). In the economic literature, only the convergence of the methods considered is proved. In contrast, this paper provides an accurate analysis of the convergence rate of the described procedures for determining the equilibrium. The analysis is based on the primal-dual nature of the algorithms proposed. More precisely, in this paper, we propose the economic interpretation of the following numerical primal-dual methods of convex optimization: dichotomy and subgradient projection method.

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Acknowledgments

The authors are grateful to A. Nedic, Yu.E. Nesterov, and A.A. Shananin for valuable comments on the paper.

Funding

This work was supported by the Russian Foundation for Basic Research, grant no. 18-29-03071, and by the Council for Grants (under RF President), grant no. MD-1320.2018.1.

Author information

Authors and Affiliations

  1. Far Eastern Federal University, ul. Sukhanova 8, Vladivostok, 690091, Russia

    E. A. Vorontsova & E. A. Nurminsky

  2. Universite de Grenoble-Alpes, Ave. Central 621, Saint-Martin-d’Heres, 38400, France

    E. A. Vorontsova

  3. Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, Moscow Region, 141700, Russia

    A. V. Gasnikov & A. S. Ivanova

  4. Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoi Karetnyi per. 19, build. 1, Moscow, 127051, Russia

    A. V. Gasnikov

  5. Caucasus Mathematical Center, Adyghe State University, ul. Pervomayskaya 208, Maykop, Republic of Adygea, 385000, Russia

    A. V. Gasnikov

Authors
  1. E. A. Vorontsova
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  2. A. V. Gasnikov
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  3. A. S. Ivanova
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  4. E. A. Nurminsky
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Corresponding authors

Correspondence to E. A. Vorontsova, A. V. Gasnikov, A. S. Ivanova or E. A. Nurminsky.

Additional information

Russian Text © The Author(s), 2019, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2019, Vol. 22, No. 4, pp. 411–432.

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Vorontsova, E.A., Gasnikov, A.V., Ivanova, A.S. et al. The Walrasian Equilibrium and Centralized Distributed Optimization in Terms of Modern Convex Optimization Methods by an Example of the Resource Allocation Problem. Numer. Analys. Appl. 12, 338–358 (2019). https://doi.org/10.1134/S1995423919040037

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  • DOI: https://doi.org/10.1134/S1995423919040037

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