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

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

The application of variational inequality theory to the study of spatial equilibrium and disequilibrium

Includes bibliographical references (p. 26-29).Supported by the National Science Foundation VPW Program. RII-880361by A. Nagurney

Parallel Computation of Large-Scale Dynamic Market Network Equilibria via Time Period Decomposition

In this paper we consider a dynamic market equilibrium problem over a finite time horizon in which a commodity is produced, consumed, traded, and inventoried over space and time. We first formulate the problem as a network equilibrium problem and derive the variational inequality formulation of the problem. We then propose a parallel decomposition algorithm which decomposes the large-scale problem into T + 1 subproblems, where T denotes the number of time periods. Each of these subproblems can then be solved simultaneously, that is, in parallel, on distinct processors. We provide computational results on linear separable problems and on nonlinear asymmetric problems when the algorithm is implemented in a serial and then in a parallel environment. The numerical results establish that the algorithm is linear in the number of time periods. This research demonstrates that this new formulation of dynamic market problems and decomposition procedure considerably expands the size of problems that are now feasible to solve

Progressive equilibration algorithms : the case of linear transaction costs

Includes bibliographical references (p. 26-27).by A. Eydeland and A. Nagurney

A bi-level programming approach for the shipper-carrier network problem

The Stackelberg game betweenshippers and carriers in an intermodal network is formulated as a bi-levelprogram. In this network, shippers make production, consumption, androuting decisions while carriers make pricing and routing decisions.The oligopolistic carrier pricing and routing problem, which comprisesthe upper level of the bi-level program, is formulated either as a nonlinearconstrained optimization problem or as a variational inequality problem,depending on whether the oligopolistic carriers choose to collude orcompete with each other in their pricing decision. The shippers\u27 decisionbehavior is defined by the spatial price equilibrium principle. Forthe spatial price equilibrium problem, which is the lower level of thebi-level program, a variational inequality formulation is used to accountfor the asymmetric interactions between flows of different commoditytypes. A sensitivity analysis-based heuristic algorithm is proposedto solve the program. An example application of the bi-level programmingapproach analyzes the behavior of two marine terminal operators. Theterminal operators are considered to be under the same Port Authority.The bi-level programming approach is then used to evaluate the PortAuthority\u27s alternative investment strategies