Abstract
Using Lemke's scheme, we give a complementary pivot algorithm for computing an equilibrium for Arrow--Debreu markets under separable, piecewise-linear concave (SPLC) utilities. Despite the polynomial parity argument on directed graphs (PPAD) completeness of this case, experiments indicate that our algorithm is practical---on randomly generated instances, the number of iterations it needs is linear in the total number of segments (i.e., pieces) in all the utility functions specified in the input. Our paper settles a number of open problems: (1) Eaves (1976) gave an LCP formulation and a Lemke-type algorithm for the linear Arrow--Debreu model. We generalize both to the SPLC case, hence settling the relevant part of his open problem. (2) Our path following algorithm for SPLC markets, together with a result of Todd (1976), gives a direct proof of membership of such markets in PPAD and settles a question of Vazirani and Yannakakis (2011). (3) We settle a question of Devanur and Kannan (2008) of obtaining a “sy...
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