Rockafellian relaxation and minimum-norm slack for the Walrasian equilibrium problem

Summary · qwen2.5:32b

The study introduces a Rockafellian relaxation method for the Walrasian equilibrium problem in exchange economies that may not achieve equilibrium, using a penalized slack variable $v$ to measure distance to the nearest feasible equilibrium. This approach ensures the model remains well-posed and identifies the minimum-norm excess demand vector as the penalty increases, illustrated through an analytically stressed Shapley-Shubik example.

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Excerpt

arXiv:2607.04717v1 Announce Type: new Abstract: We propose a Rockafellian relaxation of the Walrasian equilibrium problem for an exchange economy that may not admit one. Market clearing is slackened by a non-negative variable $v$ whose norm is penalized; the relaxation is well posed throughout. As the penalty grows, the residual converges to a vector $v^*_\infty$ of minimum norm in the feasible range of excess demand, measuring the distance to the nearest equilibrium-admitting economy. A stressed Shapley--Shubik example recovers the analytical infeasibility floor to machine