We consider Nonlinear Equilibrium (NE) for optimal allocation of limited resources. The NE is a generalization of the Walras-Wald equilibrium, which is equivalent to J. Nash equilibrium in an n-person concave game. Finding NE is equivalent to solving a variational inequality (VI) with a monotone and smooth operator on $\Omega = \Re_+^n\cross\Re_+^m$. The 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 projected pseudo-gradient (PPG) and extra pseudo-gradient (EPG) methods require $O(n^2)$ operations per step. We proved convergence, established global Q-linear rate and estimated computational complexity for both PPG and EPG methods. The methods can be viewed as pricing mechanisms for establishing economic equilibrium.
Technical Report 12_01_2012, SEOR/Math, George Mason University, Fairfax, VA, USA