The Implicit Barrier from Utility Maximization: Lightweight Interior-Point Methods for Market Equilibrium

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 tâtonnement 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.