Abstract
A two-level decomposition method for nonconvex separable optimization problems with additional local constraints of general inequality type is presented and thoroughly analyzed in the paper. The method is of primal-dual type, based on an augmentation of the Lagrange function. Previous methods of this type were in fact three-level, with adjustment of the Lagrange multipliers at one of the levels. This level is eliminated in the present approach by replacing the multipliers by a formula depending only on primal variables and Kuhn-Tucker multipliers for the local constraints. The primal variables and the Kuhn-Tucker multipliers are together the higher-level variables, which are updated simultaneously. Algorithms for this updating are proposed in the paper, together with their convergence analysis, which gives also indications on how to choose penalty coefficients of the augmented Lagrangian. Finally, numerical examples are presented.
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References
Lasdon, L. S.,Optimization Theory for Large Systems, Macmillan Company, New York, New York, 1970.
Findeisen, W., Bailey, F. N., Brdys, M., Malinowski, K., Tatjewski, P., andWozniak, A.,Control and Coordination in Hierarchical Systems, John Wiley and Sons, Chichester, England, 1980.
Bertsekas, D. P.,Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, New York, 1982.
Stephanopoulos, G., andWesterberg, A. W.,The Use of Hestenes' Method of Multipliers to Resolve Dual Gaps in Engineering System Optimization, Journal of Optimization Theory and Applications, Vol. 15, pp. 285–309, 1975.
Stoilov, E.,Method of Augmented Lagrangians in Two-Level Static Optimization, Archiwum Automatyki i Telemechaniki, Vol. 22, pp. 219–237, 1977 (in Polish).
Watanabe, N., Nishimura, J., andMatsubara, M.,Decomposition in Large-Scale System Optimization Using the Method of Multipliers, Journal of Optimization Theory and Applications, Vol. 25, pp. 183–193, 1978.
Tatjewski, P.,A Hierarchical Algorithm for Large-Scale System Optimization Problems with Duality Gaps, Modelling and Optimization, Proceedings of the 11th IFIP Conference, Copenhagen, Denmark, 1983, Edited by P. Thoft-Christensen, Springer Verlag, Berlin, Germany, pp. 662–671, 1984.
Engelmann, B., andTatjewski, P.,Accelerated Algorithms of the Augmented Interaction Balance Method for Large-Scale System Optimization and Control, Systems Analysis, Modelling, and Simulation, Vol. 3, pp. 209–226, 1986.
Tatjewski, P.,New Dual Decomposition Algorithms for Nonconvex Separable Optimization Problems, Preprints of the 4th IFAC Symposium on Large-Scale Systems Theory and Applications, Zurich, Switzerland, pp. 296–303, 1986.
Bertsekas, D. P.,Convexification Procedures and Decomposition Methods for Nonconvex Optimization Problems, Journal of Optimization Theory and Applications, Vol. 29, pp. 169–197, 1979.
Tanikawa, A., andMukai, H.,A New Technique for Nonconvex Primal-Dual Decomposition of a Large-Scale Separable Optimization Problem, IEEE Transactions on Automatic Control, Vol. 30, pp. 133–143, 1985.
Fletcher, R.,A Class of Methods for Nonlinear Programming with Termination and Convergence Properties, Integer and Nonlinear Programming, Edited by J. Abadie, North-Holland, Amsterdam, Holland, 1970.
Ortega, J. M., andRheinboldt, W. C.,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.
Dennis, J. E., Jr., andSchnabel, R. B.,Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1983.
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Communicated by D. G. Luenberger
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Tatjewski, P., Engelmann, B. Two-level primal-dual decomposition technique for large-scale nonconvex optimization problems with constraints. J Optim Theory Appl 64, 183–205 (1990). https://doi.org/10.1007/BF00940031
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DOI: https://doi.org/10.1007/BF00940031