The National Energy Modeling System (NEMS) is a large-scale mathematical model that computes equilibrium fuel prices and quantities in the U.S. energy sector and is currently in use at the U.S. Department of Energy (DOE). At present, to generate these equilibrium values, NEMS iteratively solves a sequence of linear programs and nonlinear equations. This is a nonlinear Gauss-Seidel approach to arrive at estimates of market equilibrium fuel prices and quantities. In this paper, we present existence and uniqueness results for NEMS-type models based on a nonlinear complementarity/variational inequality problem format. Also, we document mathematically, for the first time, how the inputs and the outputs for each NEMS module link together.

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0The National Energy Modeling System (NEMS) is a large-scale mathematical model that computes equilibrium fuel prices and quantities in the US. energy sector. At present, to generate these equilibrium values, NEMS sequentially solves a collection of linear programs and nonlinear equations. The NEMS solution procedure then incorporates the solutions of these linear programs and nonlinear equations in a nonlinear Gauss-Seidel approach.We describe how the current version of NEMS can be formulated as a particular nonlinear complementarity problem (NCP), thereby possibly avoiding current convergence problems. In addition, we show that the SCP format is equally valid for a more general form of NEMS. We also describe several promising approaches for solving the NCP form of NEMS based on recent Newton type methods for general NCPs. These approaches share the feature of needing to solve their direction-finding subproblems only approximately. Hence, they can effectively exploit the sparsity inherent in the NEMS NCP.

--ZusammenfassungAn Iterative Methed for Solving Partitioned Linear Equations. A new iterative scheme, using two partitions of the coefficient matrix of a given linear and non-singular system of equations A x = b, is shown to always converge to the solution. The concept of two vector spaces approaching orthogonality is quantified and used to show that the eigenvalues of the iteration matrix approach zero as the vector spaces defined by the two partitions of A approach orthogonality. Eine iterative Methode ftir die Li~sung von aufgespaltenen linearen Gleiebungssystemen. Es wird ein neues iteratives Verfahren, bei dem die Koeffizientenmatrix A eines linearen, nicht-singul/irenGteichungssystems Ax=b in zwei Teile aufgespalten wird, hergeleitet, und es wird gezeigt, dab dieses Verfahren immer zur L6sung konvergiert. Der Begriff, dab zwei Vektorr/iume sich der Orthogonalit~it annfihern, wird quantifiziert; er wird verwendet, um zu zeigen, dab sich die Eigenwerte der Iterationsmatrix Null ann/ihern, wenn sich die Vektorr~iume, durch die Zweiteilung von A bestimmt, der Orthogonatit~it ann/ihern.