Duality and Decomposition in Mathematical Programming

Abstract
The problem considered is that of obtaining solutions to large nonlinear mathematical programs by coordinated solution of smaller subproblems. If all functions in the original problem are additively separable, this can be done by finding a saddle point for the associated Lagrangian function. Coordination is then accomplished by shadow prices, with these prices chosen to solve a dual program. Characteristics of the dual program are investigated, and an algorithm is proposed in which subproblems are solved for given shadow prices. These solutions provide the value and gradient of the dual function, and this information is used to update the shadow prices so that the dual problem is brought closer to solution. Application to two classes of problems is given. The first class is one whose constraints describe a system of coupled subsystems; the second is a class of multi-item inventory problems whose decision variables may be discrete.

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