📝 SummarXiv

Abstract

We study the computation of the market equilibrium in Fisher exchange markets with divisible goods and players endowed with heterogeneous utilities. In particular, we consider the decentralized polynomial-time interior-point strategies that update \emph{only} the prices, mirroring the t\^atonnement process. The key ingredient is the \emph{implicit barrier} inherent from utility maximization, which induces unbounded demand when the goods are almost free of charge. Focusing on a ubiquitous class of utilities, we formalize this observation. A companion result suggests that no additional effort is required for computing high-order derivatives; all the necessary information is readily available when collecting the best responses. To tackle the Newton systems in the interior-point methods, we present an explicitly invertible approximation of the Hessian operator with high probability guarantees, and a scaling matrix that minimizes the condition number of the linear system. Building on these tools, we design two inexact lightweight interior-point methods. One such method has $\cO(\log(\tfrac{1}{\epsilon}))$ complexity rate. Under mild conditions, the other method achieves a non-asymptotic superlinear convergence rate. Preliminary experiments are presented to justify the capability of the proposed methods for large-scale problems. Extensions of our approach are also discussed.