Decentralized optimization for distributed-lag models of discrete systems

Abstract

The approach discussed in this paper solves a general class of optimization problems for discrete dynamic systems which include distributed lag, distributed and/or multiple pure delays, and constraints both in state and control variables. The overall system equation of this problem is described by a multidimensional nonlinear difference equation of high-order which is called the distributed-lag model. Applying Lagrange duality theory to the original problem, the dual problem is formulated, and the decomposition of the optimization process in stage is obtained. It is shown that by solving the dual problem the delay terms can be easily handled and the optimal solution to the original problem is obtained without reducing the multi-dimensional high-order system equation to a conventional larger dimensional first-order system equation. It is also shown that the dual decentralized method in this paper is easier to cope with state and control constraints than the primal method in the space of control, i.e. gradient and other techniques. The approach developed in this paper is compared with other methods using a simple example, and is applied to a combined marketing and production control problem. Some computational results are included.