Abstract
This paper provides a structural analysis of decomposition algorithms using a generalization of linear splitting methods. This technique is used to identify explicitly the essential similarities and differences between several classical algorithms. Similar concepts can be used to analyze a large class of multilevel hierarchical structures.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Takahara, Y.,A Multi-Level Structure for a Class of Dynamical Optimization Problems, Case Western Reserve University, Cleveland, Ohio, MS Thesis, 1965.
Lasdon, L. S., andSchoeffer, J. D.,A Multilevel Technique for Optimization, Proceedings of Joint Automatic Control Conference, Troy, New York, 1965.
Findeisen, W.,Parametric Optimization by Primal Method in Multilevel Systems, IEEE Transactions on Systems Science and Cybernetics, Vol. SSC-4, pp. 155–164, 1968.
Lasdon, L. S.,Optimization Theory for Large-Scale Systems, The Macmillan Company, New York, New York, 1970.
Mesarovic, M. D., Macko, D., andTakahara, Y.,Theory of Multilevel, Hierarchical Systems, Academic Press, New York, New York, 1970.
Pearson, J. B.,Dynamic Decomposition Techniques, Optimization Methods for Large Systems, Edited by D. A. Wismer, McGraw-Hill Book Company, New York, New York, 1971.
Laub, A. J.,Decentralization of Optimization Problems by Iterative Coordination, University of Minnesota, Minneapolis, Minnesota, PhD Thesis, 1974.
Singh, M. G., andTitli, A.,Closed-Loop Hierarchical Control for Nonlinear Systems Using Quasilinearization, Automatica, Vol. 11, pp. 541–546, 1975.
Cohen, G.,On an Algorithm of Decentralized Optimal Control, Journal of Mathematical Analysis and Applications, Vol. 59, pp. 242–259, 1977.
Cohen, G.,Optimization by Decomposition and Coordination: A Unified Approach, IEEE Transactions on Automatic Control, Vol. AC-23, pp. 222–232, 1978.
Sandell, N. R., Jr.,Decomposition versus Decentralization in Large-Scale Systems Theory, Proceedings of 1976 IEEE Conference on Decision and Control, Clearwater, Florida, 1976.
Looze, D. P., andSandell, N. R., Jr.,Decomposition of Linear, Decentralized Stochastic Control Problems, IFAC Workshop on Control and Management of Integrated Industrial Complexes, Toulouse, France, 1977.
Lehtomaki, N. A.,Iterative Decomposition of the Lyapunov and Riccati Equations, Massachusetts Institute of Technology, Cambridge, Massachusetts, MS Thesis, 1978.
Ortega, J. M., andRheinboldt, W. C.,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.
Kitchen, J. W., Jr.,Concerning the Convergence of Iterates to Fixed Points, Studia Mathematica, Vol. 27, pp. 247–249, 1966.
Author information
Authors and Affiliations
Additional information
Communicated by P. Varaiya
This research was supported in part by ONR Contract No. N00014-76-C-0346, in part by the US Department of Energy, Division of Electric Energy Systems, Contract No. ERDA-E(49-18)-2087 at the Massachusetts Institute of Technology, and in part by the Joint Services Electronics Program, Contract No. DAAG-29-78-C-0016.
The authors would like to thank Dr. P. Varaiya, University of California at Berkeley, and Dr. D. Bertsekas for their comments and suggestions.
Rights and permissions
About this article
Cite this article
Looze, D.P., Sandell, N.R. Analysis of decomposition algorithms via nonlinear splitting functions. J Optim Theory Appl 34, 371–382 (1981). https://doi.org/10.1007/BF00934678
Issue Date:
DOI: https://doi.org/10.1007/BF00934678