PARALLEL AND SERIAL VARIATIONAL INEQUALITY DECOMPOSITION ALGORITHMS FOR MULTICOMMODITY MARKET EQUILIBRIUM PROBLEMS Anna Nagurney and Dae-Shik Kim SCHOOL OF MANAGEMENT UNIVERSITY OF MASSACHUSETTS AMHERST, MASSACHUSETTS 01003 Summary We have applied parallel and serial variational inequality (VI) diagonal decomposition algorithms to large-scale, multicommodity market equilibrium problems. These decomposition algorithms resolve the VI problems into single commodity problems, which are then solved as quadratic programming problems. The algorithms are implemented on an IBM 3090-600E, and randomly gen- erated linear and nonlinear problems with as many as 100 markets and 12 commodities are solved. The com- putational results demonstrate that the parallel diagonal decomposition scheme is amenable to paralielization. This is the first time that multicommodity equilibrium problems of this scale and level of generality have been solved. Furthermore, this is the first study to compare the efficiencies of parallel and serial VI decomposition algorithms. Although we have selected as a prototype an equilibrium problem in economics, virtually any equilibrium problem can be formulated and studied as a variational inequality problem. Hence, our results are not limited to applications in economics and operations research. Introduction Equilibrium is a central concept in diverse problems in economics and operations research. Examples include oligopolistic market equilibrium problems in which profit-maximizing firms are engaged in the production and sale of one or more goods, spatial price equilibrium problems in which commodity supply and consumption levels and interregional shipments are to be determined, pure exchange and general economic equilibrium problems, and traffic network equilibrium problems in which users of the congested network seek routes to minimize travel costs. Applications of such models arise in energy and agricultural markets, international trade, global models, and transportation planning. Historically, equilibrium models were usually refor- mulated as optimization problems, provided that a cer- tain symmetry or integr-ability assumption held for the un- derlying functions. Convex programming algorithms could then, at least in principle, be used to compute the equilibrium pattern. Utilizing such an approach, Samuelson (1952) and Takayama and Judge (1971) in- troduced a variety of spatial price equilibrium models, and Beckmann, McGuire, and Winsten (1956) studied traffic network equilibrium models. For a survey of the role of integrability in economics, see Carey (1977). Recently it has been realized that equilibrium problems, governed by distinct equilibrium conditions, can be modeled and studied within the framework of variational inequalities. Variational inequalities (VI) had originally been introduced by Hartman and Stampac- chia (1966) as a tool for the study of partial differential equations. The discovery by Dafermos (1980) that the traffic network equilibrium conditions, as stated by Smith (1979), had the structure of a finite-dimensional VI problem opened new vistas for the development of more general, asymmetric multicommodity and multi- modal models and the design of mathematically correct and convergent algorithms. Gabay and Moulin (1980), in turn, formulated the oligopolistic equilibrium problem governed by Coumot-Nash equilibrium as a VI problem; Florian and Los (1982), spatial price equilib- rium problems; Border (1985) and Dafermos (1986a), exchange price equilibria. More recently, Zhao (1989) utilized the VI formulations of the pure exchange by guest on April 9, 2016 hpc.sagepub.com Downloaded from