A Comparison between Two Kinds of Decentralized Optimality Conditions in Nonconvex Programming

A. Ten Kate
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A. Ten Kate: Centre for Development Planning, Netherlands School of Economics

Management Science, 1972, vol. 18, issue 12, B734-B743

Abstract: Decomposable mathematical programming problems sometimes admit a certain degree of decentralization in the decision process. By an informative interplay between a central authority and a number of subdivisions one hopes to establish the overall best solution. This interplay can be organized in either of two ways: direct or indirect distribution. However, an absolute prerequisite for such procedures to be successful is the availability of an optimality criterion to the centre so as to know when to stop the iterative process. It is well known that for either of two cases the optimality conditions as derived from the Kuhn-Tucker conditions for the overall problem are not satisfactory. The present paper makes a comparison between the shortcomings of the two kinds of conditions. In the case of direct distribution this condition turns out to be necessary but unfortunately not sufficient and for indirect distribution the condition is sufficient but not necessary. The insufficiency in the direct and the absence of necessity in the indirect case, which are caused by eventual nonconvexities of the problem, are explored a bit further. A simple example is presented.

Date: 1972
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