Nonlinear Equilibrium for Resource Allocation Problems

We consider Nonlinear Equilibrium (NE) for the optimal allocation of limited resources. The NE is a generalization of Walras-Wald equilibrium, which is equivalent to J. Nash equilibrium in n-person concave games. Finding NE is equivalent to solving a variational inequality (VI) with a monotone and smooth operator on Omega = R-+(n) circle times R-+(m). Projection on Omega is a very simple procedure; therefore, our main focus is two methods for which the projection on Omega is the main operation. Both pseudo-gradient projection (PGP) and extra pseudo-gradient (EPG) methods require O(n(2)) operations per step, because in both cases the main operation per step is matrix by vector multiplication. We prove convergence, establish global Q-linear rate and estimated computational complexity for both the POP and EPG methods under various assumption on the input data. Both methods can be viewed as pricing mechanisms for establishing economic equilibrium. On the other hand, they are primal-dual decomposition methods.