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

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 $v_{\mathrm{\infty }}^{\ast }$ 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