Walrasian Equilibrium and Centralized Distributed Optimization from the point of view of Modern Convex Optimization Methods on the Example of Resource Allocation Problem

We consider the resource allocation problem and its numerical solution. The following constructions are demonstrated: 1) Walrasian price-adjustment mechanism for determining the equilibrium; 2) 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 economic literature the convergence of the considered methods is only 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 suggested algorithms. More precisely, in this article we propose the economic interpretation of the following numerical primal-dual methods of convex optimization: dichotomy and subgradient projection method. Numerical experiments conclude the paper.