Simplicial Variable Dimension Restart Algorithm to Find Economic Equilibria on the Unit Simplex Using n(n+1) Rays

摘要

A new simplicial variable dimension restart algorithm is introduced to solve the zero point or equilibrium problem on the n-dimensional unit simplex S(sup n). The number of one-dimensional sets or rays along which the algorithm can leave the arbitrarily chosen starting point is n(n + 1). In case the algorithm is applied to find a general equilibrium price vector in an exchange economy with n + 1 commodities, it generates a piecewise linear path of prices, connecting the arbitrarily chosen starting price vector with an approximate equilibrium price vector. At an equilibrium price vector the excess demand of all commodities is zero. Along the path the price of a commodity is kept equal to the initial price if the excess demand of that commodity is neither minimal nor maximal. If the excess demand of a commodity becomes maximal, the price of that commodity is increased from its initial value, and if the excess demand becomes minimal, the price is decreased from its initial value. Also a new intersection theorem on S(sup n) is derived.