Abstract
We study dual functionals which have two fundamental properties. Firstly, they have a good asymptotical behavior. Secondly, to each dual sequence of subgradients converging to zero, one can associate a primal sequence which converges to an optimal solution of the primal problem. Furthermore, minimal conditions for the convergence of the Gauss-Seidel methods are given and applied to such kinds of functionals.
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Tseng, P., andBertsekas, D. P.,Relaxation Methods for Problems with Strictly Convex Separable Costs and Linear Constraints, Mathematical Programming, Vol. 38, pp. 303–321, 1987.
Bertsekas, D. P., Hosein, P. A., andTseng, P.,Relaxation Methods for Network Flow Problems with Convex Arc Costs, SIAM Journal on Control and Optimization, Vol. 25, pp. 1219–1243, 1987.
Han, S. P.,A Successive Projection Method, Mathematical Programming, Vol. 40, pp. 1–14, 1988.
Han, S. P., andLou, G.,A Parallel Algorithm for a Class of Convex Programs, SIAM Journal on Control and Optimization, Vol. 26, pp. 344–355, 1988.
Censor, Y., andLent, A.,Optimization of log x Entropy over Linear Equality Constraints, SIAM Journal on Control and Optimization, Vol. 25, pp. 921–933, 1987.
Bertsekas, D. P., andTsitsiklis, J. N.,Parallel and Distributed Computation Numerical Methods, Prentice-Hall, Englewood Cliffs, New Jersey, 1989.
Zangwill, W. I.,Nonlinear Programming: A Unified Approach, Prentice-Hall, Englewoods Cliff, New Jersey, 1969.
Auslender, A.,Méthodes Numériques pour la Décomposition de Fonctions Differentiables, Numerische Mathematik, Vol. 18, pp. 213–223, 1971.
Auslender, A., andMartinet, B.,Méthodes de Décomposition pour la Minimization d'une Fonction sur un Espace Produit, SIAM Journal on Control, Vol. 12, pp. 635–654, 1974.
Ortega, J. M., andRheinboldt, W. C.,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.
Auslender, A., andCrouzeix, J. P.,Well-Behaved Asymptotical Convex Functions, Analyse Nonlinéaire, Edited by Attouch, Gauthiers-Villars, Paris, France, pp. 101–122, 1989.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Rockafellar, R. T.,Monotone Operators and the Proximal Point Algorithm, SIAM Journal on Control and Optimization, Vol. 14, pp. 877–898, 1976.
Tseng, P.,Applications of a Splitting Algorithm to Decomposition in Convex Programming and Variational Inequalities, Technical Report LIDS 1836, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, 1988.
Tseng, P.,Coordinate Ascent for Maximizing Nondifferentiable Concave Functions, Technical Report LIDS 1840, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, 1988.
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Communicated by O. L. Mangasarian
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Auslender, A. Asymptotic properties of the fenchel dual functional and applications to decomposition problems. J Optim Theory Appl 73, 427–449 (1992). https://doi.org/10.1007/BF00940050
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DOI: https://doi.org/10.1007/BF00940050