Existence of optimal Lagrange multipliers for certain nonconvex allocation problems

The use of Lagrange multipliers for decentralization of large resource allocation problems is well known. However, these dual techniques may suffer from the drawback ofduality gaps, to guarantee the absence of which various functions are required to be convex. This limits greatly the applicability of the decentralized approach. We show that less restrictive conditions can be formulated for a certain class of allocation problems, which we call resource management problems, which typically occur in large operational systems. We present a theorem for the existence of optimal multipliers, while placing almost no restrictions on the forms of the resource usage functions or the domains of the decision variables. Efficient solution algorithms, with provable convergence properties, have been given in a companion paper. Our results justify the application of dual methods to this class ofreal-world problems.The author is indebted to Mr. G. Karady and Professor Y. C. Ho of Harvard University for their valuable comments, and also to the referees for their helpful suggestions. This research was partially supported by the Office of Naval Research, under the Joint Services Electronic Program, Contract No. N0001475-C-0648, and by the National Science Foundation, Grant No. ENG-78-15231.