The Walrasian equilibrium and centralized distributed optimization in terms of modern convex optimization methods on the example of resource allocation problem
E.A. Vorontsova1,2, A.V. Gasnikov3,4,5, A.C. Ivanova3, E.A. Nurminsky1
1Far Eastern Federal University, Vladivostok, Russia
2Universite de Grenoble-Alpes, Saint-Martin-d'Heres, France
3Moscow Institute of Physics and Technology, Dolgoprudny, Russia
4Institute for Information Transmission Problems RAS, Moscow, Russia
5Adyghe State University, Maikop, Russia
Keywords: вальрасов механизм, децентрализация цен, прямо-двойственный метод, субградиентный метод, условие Слейтера, Walrasian equilibrium, decentralized pricing, primal-dual method, subgradient method, Slater condition
Abstract
We consider the resource allocation problem and its numerical solution. The following is demonstrated: 1) the Walrasian price-adjustment mechanism for determining the equilibrium; 2) the decentralized role of the prices; 3) Slater's method for price restrictions (dual Lagrange multipliers); 4) a new mechanism for determining equilibrium prices, in which prices are fully controlled not by Center (Government), but by economic agents -- nodes (factories). 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 the convex optimization: dichotomy and subgradient projection method.
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