A Splitting Equilibration Algorithm for the Computation of Large-Scale Constrained Matrix Problems; Theoretical Analysis and Applications

In this paper we introduce a general parallelizable computational method for solving a wide spectrum of constrained matrix problems. The constrained matrix problem is a core problem in numerous applications in economics. These include the estimation of input/output tables, trade tables, and social/national accounts, and the projection of migration flows over space and time. The constrained matrix problem, so named by Bacharach, is to compute the best possible estimate X of an unknown matrix, given some information to constrain the solution set, and requiring either that the matrix X be a minimum distance from a given matrix, or that X be a functional form of another matrix. In real-world applications, the matrix X is often very large (several hundred to several thousand rows and columns), with the resulting constrained matrix problem larger still (with the number of variables on the order of the square of the number of rows/columns; typically, in the hundreds of thousands to millions). In the classical setting, the row and column totals are known and fixed, and the individual entries nonnegative. However, in certain applications, the row and column totals need not be known a priori, but must be estimated, as well. Furthermore, additional objective and subjective inputs are often incorporated within the model to better represent the application at hand. It is the solution of this broad class of large-scale constrained matrix problems in a timely fashion that we address in this paper. The constrained matrix problem has become a standard modelling tool among researchers and practitioners in economics. Therefore, the need for a unifying, robust, and efficient computational procedure for solving constrained matrix problems is of importance. Here we introduce a.n algorithm, the splitting equilibration algorithm, for computing the entire class of constrained matrix problems. This algorithm is not only theoretically justiflid, hilt l'n fi,1 vl Pnitsf htnh thP lilnlprxing s-trlrtilre of thpCp !arop-Cspe mrnhlem nn the advantages offered by state-of-the-art computer architectures, while simultaneously enhancing the modelling flexibility. In particular, we utilize some recent results from variational inequality theory, to construct a splitting equilibration algorithm which splits the spectrum of constrained matrix problems into series of row/column equilibrium subproblems. Each such constructed subproblem, due to its special structure, can, in turn, be solved simultaneously via exact equilibration in closed form. Thus each subproblem can be allocated to a distinct processor. \We also present numerical results when the splitting equilibration algorithm is implemented in a serial, and then in a parallel environment. The algorithm is tested against another much-cited algorithm and applied to input/output tables, social accounting matrices, and migration tables. The computational results illustrate the efficacy of this approach

A parallel network equilibration algorithm for a class of constrained matrix problems

In this paper we propose a network equilibration algorithm for the solution of a class of constrained matrix problems with transportation-type constraints. The algorithm decomposes the problem into two series of supply and demand network equilibrium problems with special structure which can then be solved exactly and in parallel. The theoretical results obtained include the proof of convergence, the rate of convergence, and computational complexity analysis, and are obtained by interpreting the algorithm as a dual method. Computational results on datasets illustrate the theory and the efficiency of this approach

Modelling Electricity Swaps with Stochastic Forward Premium Models

We present a new model for pricing electricity swaps. Two general factors affect contracts but unique risk elements affect each contract. General factors are average swap prices and deterministic trend-seasonal components, and unique elements are forward premiums. Innovations follow MNIG distributions. We estimate the model with data from the European Energy Exchange. The model outperforms four competitors, both in in-sample valuation and in out-of-sample forecasting, and in fitting the term structure of volatilities by market segments. Competitor models are (i) diffusion spot prices, (ii) jump-diffusion spot prices with time dependent volatility, (iii) HJM-based and (iv) Levy multifactor model with NIG distributions. Value-at-Risk measures based on normality strongly underestimate tail risk but our model gives estimates that are more exact.Juan Ignacio Peña and Rosa Rodriguez acknowledge financial support from the Ministry of Economics and Competitiveness, respectively, through grants ECO2012-35023, ECO2016-77807-P, and ECO2012-3655