Journal of Computational Mathematics Vol.28, No.5, 2010, 569{578.

Abstract
We study competitive economy equilibrium computation. We show that, for the ¯rst
time, the equilibrium sets of the following two markets: 1. A mixed Fisher and Arrow-
Debreu market with homogeneous and log-concave utility functions; 2. The Fisher and
Arrow-Debreu markets with several classes of concave non-homogeneous utility functions;
are convex or log-convex. Furthermore, an equilibrium can be computed as convex opti-
mization by an interior-point algorithm in polynomial time.
Mathematics subject classi¯cation: 90C25, 91B50

Key words: Convex optimization, Competitive economy equilibrium, Non-homogeneous
utility