A FPTAS for Computing a Symmetric Leontief Competitive Economy Equilibrium

Authors:
Zhisu Zhu;Chuangyin Dang;Yinyu Ye
Affiliations:
Institute for Computational and Mathematical Engineering, Stanford University, Stanford 94305-4042;Department of Manufacturing Engineering & Engineering Management, City University of Hong Kong, Kowloon, Hong Kong SAR;Department of Management Science and Engineering, Stanford University, Stanford, 94305-4042
Venue:
WINE '08 Proceedings of the 4th International Workshop on Internet and Network Economics
Year:
2008

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Abstract

We consider a linear complementarity problem (LCP) arisen fromthe Arrow-Debreu-Leontief competitive economy equilibrium where theLCP coefficient matrix is symmetric. We prove that the decisionproblem, to decide whether or not there exists a complementarysolution, is NP-complete. Under certain conditions, an LCP solutionis guaranteed to exist and we present a fully polynomial-timeapproximation scheme (FPTAS) for computing such a solution,although the LCP solution set can be non-convex or non-connected.Our method is based on solving a quadratic social utilityoptimization problem (QP) and showing that a certain KKT point ofthe QP problem is an LCP solution. Then, we further show that sucha KKT point can be approximated with running time$\mathcal{O}((\frac{1}{\epsilon})\log (\frac{1}{\epsilon})\log(\log(\frac{1}{\epsilon}))$ in accuracyε ∈ (0,1) and a polynomial in problemdimensions. We also report preliminary computational results whichshow that the method is highly effective.