Computing Competitive Equilibrium for Chores: Linear Convergence and Lightweight Iteration

TL;DR

This paper introduces a novel approach for computing exact competitive equilibrium for chores with guaranteed linear convergence and a lightweight, subproblem-free algorithm that outperforms existing methods.

Contribution

It proposes a new unconstrained difference-of-convex formulation with convergence guarantees and develops efficient algorithms for exact and approximate CE computation.

Findings

First algorithm with provable linear convergence to exact CE.

Subproblem-free algorithm finds approximate CE in polynomial time.

Proposed methods outperform state-of-the-art algorithms in experiments.

Abstract

Competitive equilibrium (CE) for chores has recently attracted significant attention, with many algorithms proposed to approximately compute it. However, existing algorithms either lack iterate convergence guarantees to an exact CE or require solving high-dimensional linear or quadratic programming subproblems. This paper overcomes these issues by proposing a novel unconstrained difference-of-convex formulation, whose stationary points correspond precisely to the CE for chores. We show that the new formulation possesses the local error bound property and the Kurdyka-{\L}ojasiewicz property with an exponent of $1/2$. Consequently, we present the first algorithm whose iterates provably converge linearly to an exact CE for chores. Furthermore, by exploiting the max structure within our formulation and applying smoothing techniques, we develop a subproblem-free algorithm that finds an approximate CE in polynomial time. Numerical experiments demonstrate that the proposed algorithms outperform the state-of-the-art method.