Abstract
The auxiliary problem principle allows one to find the solution of a problem (minimization problem, saddle-point problem, etc.) by solving a sequence of auxiliary problems. There is a wide range of possible choices for these problems, so that one can give special features to them in order to make them easier to solve. We introduced this principle in Ref. 1 and showed its relevance to decomposing a problem into subproblems and to coordinating the subproblems. Here, we derive several basic or abstract algorithms, already given in Ref. 1, and we study their convergence properties in the framework of i infinite-dimensional convex programming.
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References
Arrow, K. J., andHurwicz, L.,Decentralization and Computation in Resource Allocation, Essays in Economics and Econometrics, Edited by R. W. Pfouts, University of North Carolina Press, Rayleigh, North Carolina, 1960.
Takahara, Y.,Multilevel Approach to Dynamic Optimization, Systems Research Center, Case Western Reserve University, Report No. SRC-50-C-64-18, 1964.
Lasdon, L. S., andSchoeffler, J. D.,A Multilevel Technique for Optimization, Proceedings of the Joint Automatic Control Conference, Troy, New York, 1965.
Brosilow, C. B., Lasdon, L. S., andPearson, J. D.,Feasible Optimization Methods for Interconnected Systems, Proceedings of the Joint Automatic Control Conference, Troy, New York, 1965.
Mesarovic, M. D., Macko, D., andTakahara, Y.,Theory of Hierarchical Multilevel Systems, Academic Press, New York, New York, 1970.
Cohen, G.,Optimization by Decomposition and Coordination: a Unified Approach, IEEE Transactions on Automatic Control, Special Issue on Large-Scale Systems, Vol. AC-23, No. 2, 1978.
Ekeland, I., andTemam, R.,Convex Analysis and Variational Problems, North-Holland Publishing Company, Amsterdam, Holland, 1970.
Ortega, J. M., andRheinboldt, W. C.,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Luenberger, D. G.,Optimization by Vector Space Methods, John Wiley and Sons, New York, New York, 1969.
Bertsekas, D. P.,Multiplier Methods: A Survey, Automatica, Vol. 12, pp. 133–146, 1976.
Cohen, G., andJoalland, G.,Coordination Methods by the Prediction Principle in Large Dynamic Constrained Optimization Problems, Proceedings of the IFAC Symposium on Large-Scale Systems, Edited by G. Guardabassi and A. Locatelli, Udine, Italy, 1976.
Sundareshan, M. K.,Generation of Multilevel Control and Estimation Schemes for Large-Scale Systems: A Perturbational Approach, IEEE Transactions on Systems, Man, and Cybernetics, Vol. SMC-7, No. 3, 1977.
Cohen, G.,Coordination of Constrained Optimization Problems by Resource Allocation, Ecole des Mines, Centre d'Automatique-Informatique, Internal Report No. A/49, 1972.
Bernhard, P.,Commande Optimale, Décentralisation, et Jeux Dynamiques, Dunod, Paris, France, 1975.
Geoffrion, A. M.,Primal Resource Directive Approach for Optimizing Nonlinear Decomposable Systems, Operations Research, Vol. 18, No. 3, 1970.
Gabay, D., andMercier, B.,A Dual Algorithm for the Solution of a Nonlinear Variational Problem via Finite-Element Approximation, Computers and Mathematics with Applications, Vol. 2, pp. 17–40, 1976.
Joalland, G., andCohen, G.,Optimal Control of a Water Distribution Network by Two Multilevel Methods, Proceedings of the 7th IFAC World Congress, Helsinki, Finland, Edited by A. Niemi, Pergamon Press, London, England, 1978.
Bensoussan, A., Lions, J. L., andTemam, R.,Sur les Méthodes de Décomposition, de Décentralisation, et de Coordination et Applications, Méthodes Numériques d'Analyse de Systèmes, Vol. 2, Cahier IRIA No. 11, 1972.
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Communicated by D. Q. Mayne
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Cohen, G. Auxiliary problem principle and decomposition of optimization problems. J Optim Theory Appl 32, 277–305 (1980). https://doi.org/10.1007/BF00934554
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DOI: https://doi.org/10.1007/BF00934554