A FPTAS for Computing a Symmetric Leontief Competitive Economy Equilibrium

Abstract

We consider a linear complementarity problem (LCP) arisen from the Arrow-Debreu-Leontief competitive economy equilibrium where the LCP coefficient matrix is symmetric. We prove that the decision problem, to decide whether or not there exists a complementary solution, is NP-complete. Under certain conditions, an LCP solution is guaranteed to exist and we present a fully polynomial-time approximation 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 utility optimization problem (QP) and showing that a certain KKT point of the QP problem is an LCP solution. Then, we further show that such a KKT point can be approximated with running time $\mathcal{O}((\frac{1}{\epsilon})\log (\frac{1}{\epsilon})\log( \log(\frac{1}{\epsilon}))$ in accuracy ε ∈ (0,1) and a polynomial in problem dimensions. We also report preliminary computational results which show that the method is highly effective.