Decomposition Methods by Prices

Abstract

In the preceding chapter (section 9.7) we showed how general d.c. algorithms can be specialized to produce decomposition methods for convex programs with an additional monotonic reverse convex constraint. In this chapter we extend two classical decomposition methods for convex programming to reverse convex programming. The first method is the price-directive approach represented by Dantzig-Wolfe’s column generation method and the second is the resource-directive approach represented by Benders’ partitioning method. Both methods are important not only for computational purpose but also for the conceptual analysis of hierarchical planning. In fact, they can be interpreted as coordination mechanisms for the decentralization of large-scale organizations (see, e.g., Dantzig (1963), Lasdon (1970), Dirickx and Jennergren (1979). In the price-directive decomposition the center issues prices for the utilization of joint resources and the subsystems suggest the sizes of their activities based on these prices. In the resource-directive decomposition the center specifies the distribution of common production factors and this distribution is determined iteratively on the basis of price-information from the subsystems.