Parallel Computation of Large-Scale Nonlinear Network Problems in the Social and Economic Sciences

In this paper we focus on the parallel computation of large - scale equilibrium and optimization problems arising in the social and economic sciences. In particular, we consider problems which can be visualized and conceptualized as nonlinear network flow problems. The underlying network structure is then exploited in the development of parallel decomposition algorithms. We first consider market equilibrium problems, both dynamic and static, which are formulated as variational inequality problems, and for which we propose parallel decomposition algorithms by time period and by commodity, respectively. We then turn to the parallel computation of large-scale constrained matrix problems which are formulated as optimization problems and discuss the results of parallel decomposition by row/column

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

Serial and parallel equilibration of large-scale constrained matrix problems with application to the social and economic sciences

We have considered the computation of large-scale con strained matrix problems, which arise in numerous ap plications in the social and economic sciences—in par ticular, the general quadratic problem that permits alter native weighting mechanisms on the data and includes the much-studied diagonal problem. The procedure uti lized is row equilibration, column equilibration (RC), which exploits the bipartite network structure of the problem by decomposing it into simpler subproblems that can be solved exactly and in parallel. We compared the efficiency of the RC algorithm to that of a well- known algorithm and established that RC was faster. We used serial RC as a benchmark for our parallel experi mentation and investigated its absolute efficiency on economic data sets and on very large quadratic diagonal problems. We implemented the RC algorithm using Par allel Fortran Prototype on the IBM 3090-600E. The results demonstrate that a constrained matrix problem with as many as a million variables can be solved using RC in minutes of CPU time in a serial environment. Speedups with the parallelized RC algorithm were substantial for the diagonal problems and moderate for the general problems. These computational results broaden the po tential domain of constrained matrix applications

Parallel computation of large-scale network equilibria and variational inequalities.

Equilibrium of a network is obtained when each user who competes to optimize his utility can not improve his utility any further. Equilibrium problems governed by distinct equilibrium concepts can be formulated in one general framework--that of variational inequalities. The synthesis of variational inequalities and networks induces the creation of highly efficient algorithms which are especially suited for the large-scale equilibrium problems. Motivated by the recent technological advances in parallel computing architectures, parallel algorithms of large-scale equilibrium problems were developed using the theory of variational inequalities. In the case where the feasible constraint set of a network equilibrium problem can be expressed as a Cartesian product of subsets, the application of variational inequality decomposition algorithms for the parallel computation becomes possible. A new spatial price equilibrium model, which is not based on the path flows, but, rather, on the link flows to allow the decomposition by time periods, was developed and used as a prototype of large-scale network equilibrium problems. The variational inequality formulations were decomposed first by commodities, then by time periods, and, subsequently, by markets. The coarse grain parallel architectures used were the IBM 3090-600E and the IBM 3090-600J at the Cornell Theory Center with six processors each. The maximum speed-ups obtained were 1.93 for two processors, 3.74 for four processors, and 5.15 for six processors. The market subproblems were further decomposed by links, resulting in a fine grain parallel implementation. The Thinking Machine\u27s Connection Machine, CM-2, with 32,768 processors was used for the numerical experimentation. The fine grain parallel algorithm solved input/output matrix problems more than 20 times faster, when compared to the results on the IBM 3090-600J. It is expected that further enhancements to parallel languages and parallel architectures will make even more efficient implementations realizable, and that parallel computing and the theory of variational inequalities can be successfully applied to solve more efficiently other large-scale problems with an underlying network structure, such as traffic equilibrium problems, general economic equilibrium problems, and financial equilibrium problems

Indefinite Knapsack Separable Quadratic Programming: Methods and Applications

Quadratic programming (QP) has received significant consideration due to an extensive list of applications. Although polynomial time algorithms for the convex case have been developed, the solution of large scale QPs is challenging due to the computer memory and speed limitations. Moreover, if the QP is nonconvex or includes integer variables, the problem is NP-hard. Therefore, no known algorithm can solve such QPs efficiently. Alternatively, row-aggregation and diagonalization techniques have been developed to solve QP by a sub-problem, knapsack separable QP (KSQP), which has a separable objective function and is constrained by a single knapsack linear constraint and box constraints. KSQP can therefore be considered as a fundamental building-block to solve the general QP and is an important class of problems for research. For the convex KSQP, linear time algorithms are available. However, if some quadratic terms or even only one term is negative in KSQP, the problem is known to be NP-hard, i.e. it is notoriously difficult to solve. The main objective of this dissertation is to develop efficient algorithms to solve general KSQP. Thus, the contributions of this dissertation are five-fold. First, this dissertation includes comprehensive literature review for convex and nonconvex KSQP by considering their computational efficiencies and theoretical complexities. Second, a new algorithm with quadratic time worst-case complexity is developed to globally solve the nonconvex KSQP, having open box constraints. Third, the latter global solver is utilized to develop a new bounding algorithm for general KSQP. Fourth, another new algorithm is developed to find a bound for general KSQP in linear time complexity. Fifth, a list of comprehensive applications for convex KSQP is introduced, and direct applications of indefinite KSQP are described and tested with our newly developed methods. Experiments are conducted to compare the performance of the developed algorithms with that of local, global, and commercial solvers such as IBM CPLEX using randomly generated problems in the context of certain applications. The experimental results show that our proposed methods are superior in speed as well as in the quality of solutions